r, p aromaticity and anti-aromaticity as retrieved by the

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Cite this: Phys. Chem. Chem. Phys., 2013, 15, 2882

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r, p aromaticity and anti-aromaticity as retrieved by the linear response kernel† Stijn Fias,*a Paul Geerlings,a Paul Ayersb and Frank De Proft*a The chemical importance of the linear response kernel from conceptual Density Functional Theory (DFT) is investigated for some s and p aromatic and anti-aromatic systems. The effect of the ring size is studied by looking at some well known aromatic and anti-aromatic molecules of different sizes, showing that the linear response is capable of correctly classifying and quantifying the aromaticity for five- to eight-membered aromatic and anti-aromatic molecules. The splitting of the linear response in s and p contributions is introduced and its significance is illustrated using some s-aromatic molecules. The linear response also correctly predicts the aromatic transition states of the Diels–Alder reaction and the acetylene trimerisation and shows the expected behavior along the reaction coordinate, proving

Received 12th October 2012, Accepted 17th December 2012

that the method is accurate not only at the minimum of the potential energy surface, but also in non-

DOI: 10.1039/c2cp43612d

Delocalisation Index (PDI), found in previous and the present study, is proven mathematically. These

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results show the linear response to be a reliable DFT-index to probe the s and p aromaticity or antiaromaticity of a broad range of molecules.

equilibrium states. Finally, the reason for the close correlation between the linear response and the Para

1 Introduction Aromaticity is a key concept in physical organic and inorganic chemistry, it has been shown to be a very useful quantity in the rationalisation of structure, stability and reactivity of many molecules.1,2 It is associated with the cyclic delocalisation of electrons, resulting in extra stabilisation in the case of aromatic compounds and destabilisation in the case of antiaromatic compounds, but aromaticity is not by itself a directly measurable quantity. This absence of an immediate observable has resulted in a large number of approaches to rationalise and quantify the aromaticity of a molecule. These can be roughly divided into five categories: energetic,3 structural or geometrical,4 electronic,5 magnetic6–10 and reactivity-based measures.11 Although many chemists link the aromaticity of a molecule with its reactivity, the last class of reactivity-based indices has been relatively unexplored. Some work has been preformed by Zhou, Parr and Garst12,13 and by De Proft and Geerlings14 to put forward the a

General Chemistry, Free University of Brussels (VUB), Pleinlaan 2, B-1050 Brussels, Belgium. E-mail: [email protected], [email protected]; Fax: +32 2-629-33173, +32 2-629-33173 b Department of Chemistry, McMaster University, Hamilton, Ontario L8S4M1, Canada † Electronic supplementary information (ESI) available: The complete set of linear response values, including those for the hydrogen atoms, of the molecules discussed in this paper. See DOI: 10.1039/c2cp43612d

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hardness as a measure of aromaticity, but the use of reactivity indices to describe the aromaticity has remained rather limited. During a chemical reaction both the molecule’s number of electrons (N) and external potential (i.e. the potential due to the nuclei, v(r)) can change. It can thus be expected that the system’s response to changes in N and/or v(r) will yield important chemical information about the system. The change of the electronic energy (E), dE, from one ground state to another, upon a change in electrons dN and external potential dv(r) can be expressed as: @E @E dE ¼ dN þ dvðrÞ ð1Þ @N vðrÞ @vðrÞ N where (qE/qN)v(r) was proven to be equal to the electronic chemical potential m, which is the negative of the electronegativity w.15 (qE/ qv(r))N on the other hand can be identified as the electron density of the molecule, r(r). When the chemical potential m and the electron density r(r) change from one ground state to another (a second order energy change), new interesting response functions emerge, which form the basis of conceptional Density Functional Theory (DFT).16 One of these is the chemical hardness (Z): 2 @mðrÞ @ E Z¼ ¼ ð2Þ @N vðrÞ @N 2 vðrÞ The chemical hardness is a measure of the resistance of the systems towards a change in their number of electrons, molecules

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with a large hardness being the most stable structures. As stated above, the hardness of a molecule is correlated with the aromaticity of the molecule.12–14 Recently, there has been a growing interest in the calculation and interpretation of another of these response functions, the linear response kernel (w(r,r 0 )), to describe the delocalisation of electrons in hyperconjugated and aromatic molecules.17–19 The linear response kernel (w(r,r 0 )) is defined as the second order functional derivative of the electronic energy (E) with respect to the external potential (v(r)):20 @rðrÞ @2E 0 wðr; r Þ ¼ ¼ ð3Þ @vðr0 Þ N @vðrÞ@vðr0 Þ N It thus measures the response of the electron density at a point r upon a perturbation in the external potential at r 0 . Using second order perturbation theory, the linear response kernel can, for closed shell spin-restricted calculations, be approximated by the following expression:18,21,22 0

wðr; r Þ ¼ 4

N=2 X i¼1

M X

fi ðrÞfa ðrÞfa ðr0 Þfi ðr0 Þ ei ea a¼ðN=2Þþ1

dAB ¼ 4

VB

Using eqn (4), the expression for the atom condensed linear response kernel becomes: wAB ¼ 4

N=2 X i¼1

M X

1 e ea i a¼ðN=2Þþ1

Z VA

fi ðrÞfa ðrÞdr

Z VB

fa ðr0 Þfi ðr0 Þdr0 ð6Þ

¼4

N=2 X i¼1

M X

1 SA SB ea ia ai e a¼ðN=2Þþ1 i

where SAia and SBai are the overlap matrix elements for atoms A and B respectively. Several atomic partition schemes can be used to calculate the overlap between the molecular orbitals, such as Bader’s Quantum Chemical Topology Analysis (QCTA) (also know as the ‘‘Atoms in Molecules’’ method),23–25 the Hirshfeld population analysis26 or its recent adaptation, the iterative Hirshfeld method.27,28 Calculation of the atom condensed linear response matrices for a series of linear and (poly)cyclic hydrocarbons has shown that the linear response kernel is able to differentiate between inductive, resonance and hyperconjugation effects. The condensed linear response values between two atoms drop exponentially with the internuclear distance when the transfer of the atomic responses due to perturbations in the external potential takes place according to the inductive effect. The mesomeric effect leads to alternating high and low linear response values, even at large internuclear distances, especially when the corresponding atoms

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N=2 X N=2 X

SijA SjiB

ð7Þ

i¼1 j¼1

ð4Þ

where the indices i and a run over the N/2 occupied and (M N)/2 unoccupied molecular orbitals respectively. The e denote the orbital energies. By integrating r over the domain of an atom A and r 0 over the domain of an atom B, one obtains an atom condensed version of the linear response function (wAB): Z Z wðr; r0 Þdrdr0 ð5Þ wAB ¼ VA

play an active role in the possible resonance structures. The impact of hyperconjugation is significantly smaller and shows characteristics intermediate to the inductive and resonance effects.17–19 In a recent study, the link between the linear response function and the concept of aromaticity was examined for some cyclic organic molecules.19 Within that study, the Para Linear Response index (PLR), the average of the linear response values between the para positions in a six-membered ring, was introduced and calculated for some classical aromatic systems (i.e., polycyclic aromatic hydrocarbons such as benzene, naphthalene, anthracene,. . .) and some six-membered rings with heteroatoms (borazine, boroxine,. . .). These results led to the conclusion that the PLR strongly correlates with a more commonly used index to describe the aromaticity, the Para Delocalisation Index (PDI). The delocalisation index between atoms A and B, dAB, is defined as:

The para delocalisation index of a six membered ring is calculated by taking the average value of the delocalisation index between the three atom-pairs in para position to each other. The strong correlation between the PLR and the PDI indicates that aromatic trends are equally well described using both indices, making the PLR an interesting reactivity index for six-membered rings.19 Until now, only the total value of the linear response kernel has been used to study the aromaticity of molecules. We now propose to write the linear response as a sum of the contributions from different (occupied) MO’s, wAB i wAB ¼4 i

M X

1 A B Sia Sai e e i a a¼ðN=2Þþ1

ð8Þ

The different wAB i ’s can be recollected in s and p parts of the linear response. Due to symmetry, only the overlap between s (p) occupied and s (p) unoccupied MO’s will contribute to AB wAB s (wp ). The cross terms will be zero as long as the atomic basins are symmetric around the molecular plane, which has to be the case for planar molecules in any sensible atom partitioning method. In the present paper we further investigate the linear response as a measure of aromaticity. First the effect of the ring size will be studied by looking at some well known aromatic and anti-aromatic molecules of different sizes. In a second part, the linear response of some s aromatic molecules is presented to illustrate the use of the s and p splitting of the linear response. The performance of the linear response at the transition state and along the reaction path is examined in a third part by means of the Diels–Alder reaction and the acetylene trimerisation. Finally we will present a mathematical argument explaining the close correlation between the linear response and the PDI. Some of the molecules studied in the present work can also be found in the test set introduced by Feixas et al., designed to evaluate the accuracy of aromaticity indices.29


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2 Computational methods

Table 2 Linear response of C5H5, C7H7+ and C8H82+, together with the s and p contributions

All molecules were optimised at the B3LYP/6-311++G** level of theory using Gaussian09.30 The linear response was calculated using the iterative Hirshfeld atom condensation scheme27,28 using a in-house program. The linear response was calculated at the same B3LYP/6-311++G** level as the optimisation using eqn (6).

3 Results and discussion 3.1 r and p contributions to the linear response for four- to eight-membered aromatic and anti-aromatic molecules In this section we will present the analysis of the linear response function of some aromatic and anti-aromatic molecules, together with their s and p contributions. Let us first consider the archetype of an aromatic molecule, benzene, for which the values of the linear response, together with the s and p contributions, are shown in Table 1. From these values, it can be seen that the typical increase of the linear response at the para position is, as might be expected, from the p contribution alone. The s contribution decreases monotonically from the ortho to the para position, showing a minimum at the para position. The s linear response thus has the same shape as cyclohexane studied previously.18 This result was expected, since the linear response is a measure of the response of the (s or p) electron density in the domain of atom A upon perturbation of the external potential in the domain of atom B. Due to the localised nature of the s bonds, the perturbation is not transmitted through the s framework, but through the p-electron density alone. To analyse the performance of the linear response for aromatic molecules of different sizes, the values were calculated for the cyclopentadienyl anion (C5H5), the tropylium ion (cycloheptatrienyl cation, C7H7+) and the cyclooctatetraenyl dication (C8H82+), all of them are aromatic molecules with 6p electrons. The values for these molecules are collected in Table 2. C7H7+and C8H82+show the typical ‘‘zig-zag’’ shape of an aromatic molecule, with maxima in the w14/w15 and w15 positions respectively. C5H5 also shows this ‘‘aromatic’’ shape in the p contribution. The s contribution decreases monotonically for all molecules, as it does in benzene,

Table 1

wAB s+p wAB s wAB p

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Linear response of benzene, together with the s and p contributions

w11

w12

w13

w14

w15

w16

2.668 1.128 1.540

0.537 0.226 0.311

0.281 0.074 0.206

0.503 0.020 0.483

0.281 0.074 0.206

0.537 0.226 0.311


w11

w12

w13

w14

w15

w16

w17

C5H5 wAB 2.657 s+p wAB 1.181 s 1.476 wAB p

0.564 0.243 0.321

0.481 0.083 0.398

0.481 0.083 0.398

0.564 0.243 0.321

C7H7+ wAB 2.848 s+p wAB 1.143 s wAB 1.705 p

0.497 0.234 0.263

0.321 0.081 0.240

0.352 0.011 0.341

0.352 0.011 0.341

0.321 0.081 0.240

0.497 0.234 0.263

C8H82+ wAB 3.090 s+p wAB 1.184 s 1.906 wAB p

0.432 0.254 0.178

0.458 0.088 0.370

0.191 0.011 0.180

0.443 0.003 0.440

0.191 0.011 0.180

0.458 0.088 0.370

w18

0.432 0.254 0.178

indicating a localised s framework. When looking at the total values, wAB, of C8H82+, the the highest value of the linear response is w13, due to the s contribution to the total value. However, when looking at the p contribution alone, wp15 has the largest value. Because the ring size of the molecule increases, while the number of p electrons remains the same, one expects the following order in the aromaticity of the systems: C5H5 > C6H6 > C7H7+ > C8H82+.7,29 When looking at the w14 and w15 values of C7H7+and the w15 value of C8H82+, the linear response clearly classifies both molecules lower in aromaticity than benzene, but the exact position of C7H7+cannot be determined by these values. After all, due to the seven membered ring, the perturbation of the external potential in the domain of atom 1 causes an equal, but shared response in the domains of atoms 4 and 5, making it difficult to use this value as a reference. A similar problem occurs in C5H5, where the w13 and w14 values predict the molecule to be slightly less aromatic then benzene, whereas the NICS value predicts the molecule to be more aromatic than benzene.7 This can also be attributed to the shared response in the domains of atoms 3 and 4. The w(p)12 value might serve as a candidate to classify the relative order of aromaticity in this case, since it measures the ease at which the perturbation in an atomic domain is transmitted to its closest neighbor. When looking at these w12 values, the same ordering as with the NICS is found for these molecules. Cyclobutadiene (D2h) and cyclooctatetraene (D4h) were also calculated as examples of anti-aromatic molecules. To be able to perform the s/p-splitting of the linear response, cyclooctatetraene was kept in the planar D4h symmetry, which is the transition state for the tub-to-tub inversion. Their values are collected in Table 3. The double bonds in the molecules are between atoms C1QC2; C3QC4; etc. In both molecules, the highest linear response is between the atoms in the double bonds. The wAB values between atoms on the opposite sides of the ring (w13 and w15 for C4H4 and C8H8 respectively) are smaller than the response with the neighbors of these atoms (w12/w14 and w14/w16 respectively). The pattern of the linear response of these anti-aromatic molecules is almost opposite to that of the aromatic ones. They also possess a ‘‘zig-zag’’ shape in the response, but with a local minimum instead of a maximum at the atom on the opposite side of the ring.

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Table 3 Linear response of C4H4 and C8H8, together with the s and p contributions. The shorter (double) bonds are located between atoms 1 and 2 in both molecules

Table 4

Linear response of hexaiodobenzene and its dication

C6I6


w11

w12

w13

w14

C4H4 wAB 3.182 s+p wAB 1.357 s 1.825 wAB p

1.947 0.201 1.746

0.124 0.140 0.016

0.582 0.512 0.070

C8H8 wAB 2.866 s+p wAB 1.188 s wAB 1.677 p

1.290 0.172 1.118

0.139 0.086 0.053

0.486 0.010 0.476

w15

0.038 0.003 0.035

w16

0.148 0.013 0.136

w17

0.139 0.086 0.053

w18

0.080 0.299 0.218

w11

w12

w13

w14

w15

w16

wAB s+p wAB s wAB p

3.513 1.877 1.637

0.349 0.097 0.252

0.189 0.038 0.151

0.394 0.003 0.392

0.189 0.038 0.151

0.349 0.097 0.252

I6 wAB s+p wAB s wAB p

2.831 2.231 0.600

0.287 0.239 0.048

0.078 0.072 0.006

0.054 0.001 0.053

0.078 0.072 0.006

0.287 0.239 0.048

w13

w14

w15

w16

C6

C6I62+

From these and previous findings, one can postulate the following ‘‘rules’’ to identify a molecule as aromatic or anti-aromatic using the linear response:17–19 Aromatic molecules have a typical ‘‘zig-zag’’ shape, the highest linear response value between atoms on the opposite side of the aromatic ring. For benzene and other aromatic molecules with a 4n + 2 ring, the reason for this highest linear response value between atoms on the opposite side of the ring can be understood in terms of the resonance structures arising from electron pair shifts. As shown previously by Sablon et al., the atoms on opposite sides of the ring are ‘‘mesomerically active’’, explaining the higher response in this position.17–19 Anti-aromatic molecules also have a ‘‘zig-zag’’ shape, but with a local minimum between atoms on the opposite side of the aromatic ring. They have their highest value between neighboring atoms on a double bond. The explanation can also be found by looking at the resonance structures arising from electron pair shifts, as shown in Fig. 1 for cyclooctatetraene. The atoms in the 2 and 4 position are ‘‘mesomerically active’’, explaining the higher response in these positions. 3.2

The linear response of some r aromatic molecules

Recently, one of the authors studied the s-aromaticity in hexaiodobenzene and its dication using the Multi Center Bond Index (MCBI) and Ring Current Maps (RCM).31

Fig. 1

C6

w11

w12

wAB s+p wAB s wAB p

3.387 1.790 1.597

0.322 0.091 0.231

0.189 0.039 0.150

0.385 0.005 0.380

0.1894 0.039 0.150

I6 wAB s+p wAB s wAB p

8.056 7.443 0.613

0.082 0.129 0.048

1.833 1.828 0.005

2.623 2.569 0.054

1.833 1.828 0.005

0.322 0.091 0.231

0.082 0.129 0.048

The dication of this molecule shows both the traditional p-aromaticity in the C6 ring, combined with a s-aromatic delocalisation and ring current along the six iodine atoms. The neutral molecule however loses the s-aromaticity in the iodine ring. The wAB values of both molecules were calculated and the values are collected in Table 4. These data show that the linear response values of the C6-ring in hexaiodobenzene and its dication are similar, but slightly smaller than the values of benzene. The wAB s values of the iodine ring in the neutral molecule show the same monotonic decrease as in benzene, indicating a classical localised s framework. The wAB s values of the iodine ring in the dication on the other hand show a maximum at w14, typical for an aromatic molecule. Another example of a molecule exhibiting s-aromaticity is the 3,5-dehydrophenyl cation (C6H3+), (Fig. 2a), a planar molecule where the p-orbitals on the alternating dehydrogenated carbons of the six-membered ring overlap to form a threecentre, two-electron (3c–2e) s-bond in the ring plane.32,33 The p system remains similar to the one of benzene, which is why this molecule is ‘‘double aromatic’’. Fukunaga et al.34 suggested the possibility of an analogous 3c–2e bond in [5.5.5]trefoilene, C9H6 (Fig. 2b), a 10p-electron structure of which the planar structure is a local minimum on the potential energy surface. The linear response matrix of both molecules is given in Tables 5 and 6.

Resonance structures of cyclooctatetraene.

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The 3,5-dehydrophenyl cation, C6H3+, (a) and [5.5.5]trefoilene, C9H6, (b).

Table 5 Linear response of the 3,5-dehydrophenyl cation, together with the s and p contributions

wAB s+p

C1H

C2

C3H

C4

C5H

C6

C1H C2 C3H C4 C5H C6

2.649

0.476 3.757

0.378 0.476 2.649

0.75443 0.88862 0.47609 3.75655

0.37791 0.75448 0.37788 0.47609 2.64873

0.47613 0.88868 0.75448 0.88862 0.47616 3.75678

1.175

0.169 2.301

0.140 0.169 1.175

0.296 0.743 0.169 2.301

0.140 0.296 0.140 0.169 1.175

0.169 0.743 0.296 0.743 0.169 2.301

1.474

0.307 1.456

0.238 0.307 1.474

0.459 0.146 0.307 1.456

0.238 0.459 0.140 0.307 1.474

0.307 0.145 0.459 0.146 0.307 1.456

wAB s C1H C2 C3H C4 C5H C6 wAB p C1H C2 C3H C4 C5H C6

When looking at the p component of the linear response of the C6 ring in the 3,5-dehydrophenyl cation, the values are similar but slightly smaller than the ones from benzene. This is expected due to the geometrical distortion of the ring. The s component of the linear response on the other hand shows a large delocalisation between the deprotonated C atoms, clearly revealing the s–3c–2e bond between these atoms. For [5.5.5]trefoilene, the s component of the linear response once again shows the existence of a 3c–2e bond between the deprotonated C atoms. The linear response reveals this 3c–2e bond to be somewhat less aromatic compared to the one in the 3,5-dehydrophenyl cation. The interpretation of the p linear response component of [5.5.5]trefoilene, however, is less trivial. When taking carbon atom 3 as a reference, we see the ‘‘classical’’ zig-zag shape, with a maximum at w37 and w38, as expected. But when one takes one of the protonated carbon atoms as a reference, one notices that the linear response

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is maximal between the atoms related by the rotation around the C3-axis (e.g. w14 and w17), instead of being maximal on the opposite side of the ring. 3.3

The linear response along the reaction coordinate

It is well-known that the Diels–Alder reaction between ethene and butadiene goes through an aromatic transition state.29,35 To check the performance of the linear response as a measure of aromaticity the para linear response (PLR) along the reaction coordinate of this reaction was calculated. The result is given in Fig. 3. The maximum PLR is found exactly at the transition state, with a value of 0.497 au which coresponds to 98.8% of the PLR value of benzene. It is reassuring that the linear response, calculated using the approximate expression of eqn (4) yields correct results along the reaction coordinate, which proves that this expression can be used, not only at the minima of the potential energy surface, but also for transition states and along the reaction coordinate. In a previous contribution, the authors showed that the evolution of the chemical hardness along the reaction profile can be used to determine whether the cycloaddition reaction at hand is allowed or forbidden, which allowed us to restate the Woodward–Hoffmann rules for cycloaddition reactions using the language of DFT.36,37 These results prove that the cycloaddition can now also be interpreted using another DFT reactivity index, namely the linear response. This opens the possibility to also describe the well-known Woodward–Hoffmann rules within DFT using the linear response kernel. Another reaction with an aromatic transition state is the thermally allowed [2 + 2 + 2] trimerization of acetylene to benzene. In this reaction the three in-plane p-bonds of acetylene are converted in the localised s bonds of benzene and the three out-of-plane p-bonds are converted in the delocalised p orbitals of benzene. Several studies on the change of the aromaticity along the reaction path have been published.29,35,38–40 Jiao and Schleyer concluded that the transition state has a 56% s and 44% p contribution to the NICS,38 while Morao and Cossıó concluded from the height profile of the NICS that the TS had no p character at all.39

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Table 6

PCCP Linear response of [5.5.5]trefoilene, together with the s and p contributions

wAB s+p

C1H

C2H

C3

C4H

C5H

C6

C7H

C8H

C9

C1H C2H C3 C4H C5H C6 C7H C8H C9

2.917

0.241 2.917

0.190 0.348 3.742

0.424 0.283 0.349 2.919

0.306 0.424 0.190 0.242 2.919

0.386 0.386 0.727 0.190 0.348 3.741

0.424 0.306 0.387 0.424 0.284 0.348 2.919

0.283 0.424 0.386 0.306 0.424 0.190 0.242 2.919

0.348 0.190 0.727 0.386 0.387 0.727 0.190 0.349 3.741

1.165

0.233 1.165

0.108 0.070 2.223

0.046 0.071 0.070 1.166

0.047 0.046 0.108 0.234 1.166

0.192 0.192 0.588 0.108 0.070 2.223

0.046 0.047 0.192 0.046 0.071 0.070 1.166

0.071 0.046 0.192 0.047 0.046 0.108 0.234 1.166

0.070 0.108 0.588 0.192 0.192 0.588 0.108 0.070 2.223

1.752

0.008 1.752

0.082 0.278 1.519

0.378 0.213 0.279 1.753

0.259 0.378 0.083 0.008 1.753

0.194 0.194 0.139 0.082 0.278 1.518

0.378 0.259 0.194 0.378 0.213 0.278 1.753

0.213 0.378 0.194 0.259 0.378 0.082 0.008 1.753

0.278 0.082 0.139 0.194 0.194 0.139 0.083 0.279 1.519

wAB s C1H C2H C3 C4H C5H C6 C7H C8H C9 wAB p C1H C2H C3 C4H C5H C6 C7H C8H C9

Fig. 3 The para linear response (PLR) versus the reaction coordinate for the Diels–Alder reaction (IRP in amu1/2 Bohr). The PLR is given in percentages of the PLR value of benzene.

Examination of the ring current maps along the reaction profile showed that the transition state is purely a s aromatic one, with a diamagnetic s circulation throughout the carbon framework, while the p current remained localised. Only after the relocalisation of the s electrons to form the C–C s bonds does the p ring current start to appear.40 A similar conclusion was drawn by Mandado et al. by studying the different contributions to the delocalisation index.35 This molecule thus forms an interesting test case, because the s/p splitting of the linear response could help to clarify the s or p nature of the transition state. To this end the PLR was

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calculated along the reaction coordinate of this reaction, along with ts s and p contribution. The results are given in Fig. 4. The shape of the total PLR has a local maximum around the transition state, decreases when the reaction goes forward, to increase again towards the final product. The same shape is found for the para delocalisation index, and the multi centre delocalisation index.29 When looking at the s and p contribution to the PLR, the linear response reveals the aromaticity of the transition state to be almost only s in nature. The linear response of the p system only becomes significant when the s framework is returned to a more localised state. These findings


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Fig. 4 The para linear response (PLR) versus the reaction coordinate for the acetylene trimerisation (IRP in amu1/2 Bohr). The PLR is given in percentages of the PLR value of benzene.

are completely in agreement with the results obtained from the ring current maps. This confirms the conclusions drawn from the NICS height profile by Morao and Cossıó and from the ring current maps by Havenith et al., namely that the transition state of this acetylene trimerisation has no p character at all.

which, condensed to atoms A and B, gives wAB ¼ D

Z N=2 N=2 Z 4 XX f ðrÞfi ðr0 Þdðr r0 Þdrdr0 D i¼1 j¼1 VA VB i

3.4 The relation between the linear response kernel and delocalistation indices The close correlation between the linear response and the (para) delocalisation index, found in previous and the present study,17–19 might at first glance seem rather peculiar, since the first one is calculated as a sum of both occupied and unoccupied orbitals (eqn (6)), whereas the latter is calculated as just a sum over the occupied orbitals (eqn (7)). However, the relationship between ¨ld approximation.41–43 both can be shown by invoking the Unso Within this approximation, the excitation energies ei ea are replaced by some value D, representing the average of the ionisa¨ld approximation is widely used in the tion energies. The Unso calculation of e.g. polarisabilities and dispersion forces,44 for some recent applications by some of the authors, see ref. 22 and 45. ¨ld approximation to eqn (4) gives Applying the Unso wðr; r0 Þ

N=2 M 4X X f ðrÞfa ðrÞfa ðr0 Þfi ðr0 Þ ¼ wD ðr; r0 Þ ð9Þ D i¼1 a¼ðN=2Þþ1 i

Using the resolution of the identity,46 one can write the Dirac delta function, d(r r 0 ) in complete orthonormal basis set of functions fn: dðr r0 Þ ¼

1 X

fn ðrÞfn ðr0 Þ

n¼1

¼

N=2 X j¼1

1 X

fj ðrÞfj ðr0 Þ þ

fk ðrÞfk ðr0 Þ

ð10Þ

k¼ðN=2Þþ1

Inserting eqn (10) in eqn (9) gives ! N=2 N=2 X 4X 0 0 dðr r Þ fj ðrÞfj ðr Þ fi ðrÞfi ðr0 Þ ð11Þ wD ðr; r Þ ¼ D i¼1 j¼1 0

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Z VA

Z VB

fi ðrÞfj ðrÞfj ðr0 Þfi ðr0 Þ

ð12Þ

When using non-overlapping basins for atoms A and B, which is the case when using the Quantum Chemical Topology Analysis (and also using the Hirshfeld population analysis, when atoms A and B are sufficiently far apart, as is the case in the para position), the first term of eqn (12) becomes zero, and the equation becomes N=2 N=2

wAB D ¼

4 XX A B 1 S S ¼ dAB D i¼1 j¼1 ij ji D

ð13Þ

and since D is a negative quantity, this shows why there is such a close relation between the linear response and the (para) delocalisation index.

4 Conclusions The linear response shows the expected behavior in all test cases studied in this work, showing the linear response to be a reliable index to probe the aromaticity of a molecule. The linear response is capable of correctly classifying and quantifying the aromaticity for five to eight membered aromatic and anti aromatic molecules. Moreover, splitting the linear response in its s and p contributions has shown to be a useful tool to investigate the s or p nature of the aromaticity of molecules. The linear response also correctly predicts the aromatic transition states of the Diels–Alder reaction and the acetylene trimerisation, and shows the expected behavior along the reaction coordinate, proving that the method is also accurate not only at the minimum of the potential energy surface, but also in non-equilibrium states. The reason for the close correlation between the delocalisation index and the linear response can be proven mathematically. All in all, the present paper provides further evidence for the ‘‘chemistry’’ present in the at first sight awkward kernel-type linear response function w(r,r0 ), the least explored second order conceptual DFT reactivity index.

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Acknowledgements


S.F., P.G. and F.D.P. wish to acknowledge support from the Research Foundation Flanders (FWO), the Free University of Brussels through the GOA77 grant.

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