Aromaticity and Antiaromaticity of Substituted Fulvene Derivatives: Perspectives from the Cite This: Phys

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Aromaticity and antiaromaticity of substituted fulvene derivatives: perspectives from the Cite this: Phys. Chem. Chem. Phys., 2017, 19,18635 information-theoretic approach in density functional reactivity theory

Donghai Yu,a Chunying Rong,*a Tian Lu,b Pratim K. Chattaraj,c Frank De Proft*d and Shubin Liu *ae

Even though the concept of aromaticity and antiaromaticity is extremely important and widely used, there still exist lots of controversies in the literature, which are believed to be originated from the fact that there are so many aromatic types discovered and at the same time there are many aromaticity indexes proposed. In this work, using seven series of substituted fulvene derivatives as an example and with the information-theoretic approach in density functional reactivity theory, we examine these concepts from a diﬀerent perspective. We investigate the changing patterns of Shannon entropy, Fisher information, Ghosh–Berkowitz–Parr entropy, information gain, Onicescu information energy, and relative Renyi entropy on the ring carbon atoms of these systems. Meanwhile, we also consider variation trends of four representative kinds of aromaticity indexes such as FLU, HOMA, ASE and NICS. Statistical analyses among these quantities show that with the same ring structure of the derivatives, both information-theoretic quantities and aromaticity indexes obey the same changing pattern, which are Received 25th May 2017, valid across all seven systems studied. However, cross correlations between these two sets of quantities Accepted 22nd June 2017 yield two completely opposite patterns. These ring-structure dependent correlations are in good DOI: 10.1039/c7cp03544f agreement with Hu¨ckel’s 4n + 2 rule of aromaticity and 4n rule of antiaromaticity. Our results should provide a novel and complementary viewpoint on how aromaticity and antiaromaticity should be rsc.li/pccp appreciated and categorized. More studies are in progress to further our understanding about the matter.

Mo¨bius aromaticity,10 excited state aromaticity,11 hybrid aromaticity, Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. 1. Introduction homoaromaticity,12 heteroaromaticity,13 claromaticity, three- As one of the most widely used concepts in chemistry, aromaticity dimensional aromaticity,14,15 spherical aromaticity,16 cubic is associated with the cyclic delocalization of electrons resulting aromaticity,17 octahedral aromaticity,18 s-aromaticity, d-aromaticity, in extra stability compared to their other isomers with the same multiple aromaticity,19 metalloaromaticity,20 chelatoaromaticity,21 chemical formula.1–3 Closely related to aromaticity is the concept quasiaromaticity, transition state aromaticity,22 hyperaromaticity, of antiaromaticity, where for the same chemical formula one etc.5 On the other hand, there are so many different descriptors finds extra instability instead.3–8 Despite its widespread use, proposed in the literature to characterize aromaticity. Examples of controversies and discussions about aromaticity are abundant acronyms of these descriptors include ASE, RE, ISE, AI, HOMA, the in the literature.2,9 On the one hand, so many different categories Julgindex,theJugindex,theBirdindex,PDI,FLU,MCI,Iring,ING, of aromaticity have been reported, such as Hu¨ckel aromaticity,3 INB, EDDB, AV1245, ELFp,ATI,PLR,AICD,ARCS,NICS,NICSzz, 1–3,6,8,23–31 NICSp, etc. These descriptors are sometimes incompatible to each other and in many cases have been found to produce a Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), College of Chemistry and Chemical Engineering, contradictory results; there indeed exists no single index that Hunan Normal University, Changsha, Hunan 410081, P. R. China can be used as the general measure for all kinds of aromaticity b Beijing Kein Research Center for Natural Sciences, Beijing 100022, P. R. China mentioned above. c Department of Chemistry and Center for Theoretical Studies, So where is the culprit leading to this dilemma? We often Indian Institute of Technology, Kharagpur 721302, India confuse consequences with the cause. Extra stability, an equalized d Research Group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium bond distance or a planar structure, ring current, peculiar reactivity, 4,5,7,32 e Research Computing Center, University of North Carolina, Chapel Hill, etc., are all consequences, not the cause, of aromaticity. To this North Carolina 27599-3420, USA end, aromaticity is like a big elephant; the above-mentioned

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descriptors are merely gauges of some aspects of this huge The third quantity in ITA is the Ghosh–Berkowitz–Parr (GBP) animal. In this paper, we adopt the viewpoint that the cause entropy,26,53,54 ð of aromaticity and antiaromaticity is the electron density 3 tðÞr; r redistribution in a closed structure, compared to its other SGBP ¼ krðrÞ c þ ln dt (3) 2 tTFðÞr; r isomers, leading to changes in the structure, reactivity, and other properties. which was originally obtained by Ghosh, Berkowitz and Parr from To confirm this point, in this work, we tackle the issue of transcribing the ground-state density functional theory into a local aromaticity and antiaromaticity from a diﬀerent perspective. thermodynamics through a phase-space distribution function. In The tool we will use is a relatively new approach that we recently eqn (3), t(r,r) is the kinetic energy density, tTF(r;r)istheThomas– developed, density functional reactivity theory (DFRT),33,34 in Fermi kinetic energy density, and c and k are constants. Other ITA 34,55 which we employ simple electron density functionals to quantities are the Onicescu information energy of order n: ð appreciate and quantify the structure and reactivity properties 1 E ¼ rnðrÞdt; (4) of molecular systems. In particular, the information-theoretic n n 1 approach (ITA) will be utilized,35,36 where these simple density Z functionals are based on the quantities from information with n 2. Onicescu introduced this quantity in an attempt to theory.34 We choose substituted fulvene derivatives as the define a finer measure of dispersion distribution than that of ´ model systems in this study, because it has been shown earlier Shannon entropy, which is closely related to Renyi entropy and that depending on the nature of the substituting group, these Tsallis entropy. The relative Shannon entropy, also called species can be either aromatic or antiaromatic.37–40 We select information gain, Kullback–Leibler divergence, or information divergence, is defined by34 four representative descriptors from the above list probing ð aromaticity through the energetic (ASE), electron delocalization rðrÞ IG ¼ rðrÞ ln dt; (5) (FLU), geometric (HOMA), and magnetic (NICS) criteria, in this r0ðrÞ study.1,3,8,24,41 Our purpose is to examine the performance of where r (r) is the reference state density satisfying the same these descriptors together with that of the simple density 0 normalization condition as r(r). This reference density can be functionals from ITA for fulvene derivatives. Different trends from the same molecule with different conformation or from and correlations among these quantities will provide clues the reactant of a chemical reaction when the transition state is about and insights into the nature of aromaticity and anti- investigated.47 Related to information gain is the relative Re´nyi aromaticity. This work does not intend to resolve all the issues entropy of order n, which is defined as follows34,56 associated with the matter. Rather, it is our hope that it opens ð n up a new door for us to appreciate and apprehend aromaticity r 1 r ðrÞ Rn ¼ n1 dt: (6) and antiaromaticity from a new theoretical foundation. n 1 r0 ðrÞ There are two other representations for ITA quantities defined 2. Theoretical framework in eqn (1)–(6), using the shape function and the atoms-in- molecules schemes.57 More details are available in a recent According to DFT, all the structure and reactivity properties of a review.34 This approach has been applied to appreciate and

Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. molecular system in the ground state are dictated by its quantify a number of physicochemical properties. We have 42 electron density, r(r). The endeavor of employing simple shown that Fisher information is a reliable descriptor of the density functionals to quantify this dictation is what DFRT is steric effect40 and stereoselectivity.45,47 The relative Shannon 43,44 45–47 trying to accomplish. The steric eﬀect, electrophilicity/ entropy is an effective quantifier to determine electrophilicity, 48 49,50 nucleophilicity, acidity/basicity, regioselectivity/stereo- nucleophilicity, and regioselectivity.46 These quantities are 46,51 selectivity are among recent examples of this ongoing additionally applied to accurately predict molecular acidity,49,50 the 34 effort. Simple density functionals from information theory HOMO/LUMO gap,58 etc. We have also applied these quantities to 33 have been featured in a number of recent applications. Major appreciate chemical phenomena in a number of other systems.34,53 quantities from ITA include the Shannon entropy52 ð Meanwhile, to quantify aromaticity and antiaromaticity, numerous descriptors are available in the literature. Here, S ¼ rðrÞ ln rðrÞdt; (1) S we consider only four representative descriptors. The HOMA (harmonic oscillator model of aromaticity) index is a geometrical which is a local functional of the electron density. It measures measurement of the equalization of chemical bonds on an aromatic the spatial delocalization of the electron density. Fisher 7 15,52 ring. Its definition is the following, information, X ð a 2 2 HOMA ¼ 1 R R ¼ 1 jjrrðrÞ n opt i I ¼ dt; (2) F rðrÞ hiX (7) 2 a 2 a Ropt Rav þ ðÞRav Ri ; which is a functional of both the electron density and its n

gradient |rr|. Contrary to Shannon entropy, it is a gauge of where n is the bond number of the considered ring, Rav is the

the sharpness or localization of the electron density distribution. average bond length, Ri is the bond length (all in Å), of each

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bond on the ring, and a is the normalization constant (for C–C 3. Computational details bonds a = 257.7). A HOMA value of o0, =0, and =1 represents antiaromatic, nonaromatic, and perfect aromatic systems, Scheme 1 exhibits the systems studied in this work, including respectively.59 As an electronic descriptor, the aromatic fluctuation tri (3MR), tetra (4MR+ and 4MR with one positive and index (FLU) describes the fluctuation of electronic charge between negative charge, respectively), penta (5MR), hexa (6MR+ and adjacent atoms in a given ring. Its definition is as follows,26 6MR, with one positive and negative charge, respectively), and hepta- (7MR) fulvene derivatives. The substituting group R "# Xring d 2 was chosen from the following pool, R = H, CH3, CCH, CMe3, 1 FluðA ! BÞ dðA; BÞdref ðA; BÞ FLU ¼ ; (8) CN, CONH2, COCH3,CF3,CH2 ,CC , COO , F, B(OH)2, OH, n FluðB ! AÞ dref ðA; BÞ + + AB OCH3,O ,NH2,NO2, NO, NMe2,NH ,NH3 ,NN, and SiMe3, with the criterion that the optimized structure should be planar. dðA; BÞ dðA; BÞ All calculations were performed at the DFT B3LYP/6-311G(d,p) with FluðA ! BÞ¼ P ¼ , dðA; BÞ 2½NðAÞlðAÞ level of theory60–62 using the Gaussian 09 package version E0163 a ( B A with the tight SCF convergence criterion and ultra-fine integration 1FluðA!BÞ4FluðB!AÞ NP=2 grids. A single-point frequency calculation was followed to ensure d¼ ; and dðA;BÞ¼4 Si;jðAÞSi;jðBÞ, 1FluðA!BÞFluðB!AÞ i;j that the final structure obtained has no imaginary frequency.

where Sij(A) is the overlap of the molecular orbitals i and j within The MultiWFN 3.3.8 program developed by one of the present the basin of atom A. The smaller the FLU value, the stronger the authors was used to calculate the information-theoretic quantities aromaticity. The third descriptor is the nucleus-independent introduced above with the check point file generated from the 64 chemical shift (NICS) derived from the effect of aromatic ring above Gaussian calculations as the input file. To obtain current.6,8 It is found that diamagnetic or diatropic ring current the electron density for the isolated state, we employed the is associated with aromaticity whereas a paramagnetic or para- spherically-averaged electron density of the neutral atom at the tropic ring current signals antiaromaticity. This difference in same level of theory. To perform the atomic partition, Becke’s 65 ring current leads to a noticeable difference in NMR chemical fuzzy atom approach, Bader’s zero-flux atoms-in-molecules 66 67 shifts.6,27 A more negative value of NICS is an indication of a criterion, and Hirshfeld’s stockholder approach are possible. 60 stronger aromaticity, and vice versa. In formulation, NICS at the As has been demonstrated earlier, these three approaches yield chosen points located at or on top of the aromatic ring can be qualitatively similar results. In this work, we choose Hirshfeld’s described as the sum of partial chemical shifts arising from stockholder approach to partition atoms in molecules to obtain 8,41 occupied molecular orbitals Ck0. atomic values of above IT quantities on carbons atoms in the *+ *+ aromatic/antiaromatic ring. The NICS(1) aromaticity index was Xocc Xocc 39 1 rrN I rN r 2 ðÞLN calculated using the procedure from the literature. FLU and r ¼ C C C C 64 2c2 k0 3 k0 c k0 3 k1 HOMA indexes were obtained from the MultiWFN package. The k jjr RN k jjr RN ASE index for 3MR–7MR species in Scheme 1 was computed as (9) the total energy difference from the isodesmic reaction shown in Scheme 2 with the same level of theory. Unit for ASE is kJ mol1, where the first and second terms on the right-hand side of for NICS is ppm, and for IT quantities atomic units.

Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. eqn (9) are the diamagnetic and paramagnetic contributions,

respectively, LN = rN rrefers to the angular momentum operator, rN = r RN, r is the electron position, and RN is the 4. Results and discussion chosen points for the calculation. In this study, we chose NICS(1), which means that the chosen point is 1 Å above the As an example, Table 1 shows the numerical results of the four aromatic ring. Other NICS indexes yield qualitatively similar aromaticity indexes for two substituted fulvene derivatives, 3MR results,6,8 giving the same trend for fulvene derivatives.39 and 6MR. As can be seen from Table 1, in good agreement with The last descriptor ASE (aromatic stabilization energies) that the previous findings, these derivatives can be both aromatic and we considered here is an energetics index. It is the total energy antiaromatic.37,39,40 For instance, using ASE as the aromaticity difference of an isodesmic or homodesmotic reaction or iso- criterion, for 3MR, a majority of the 3MR derivatives are aromatic, merization between an aromatic structure and its other structures with ASE o 0, while most of the 6MR+ species are antiaromatic, with the same chemical formula. A negative value of ASE denotes with ASE 4 0. This changing pattern between aromaticity and the existence of aromaticity, whereas a positive ASE value suggests antiaromaticity is also recovered from the numerical value of that the system is of antiaromatic nature.1,3 HOMA, where, as shown from the Table 1, for 3MR, its values are

Scheme 1 Substituted fulvene derivatives to be studied in this work.

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Scheme 2 Isomerization reactions to obtain the aromatic stabilization energy (ASE) for substituted fulvene derivatives 3MR–7MR.

often small, but for 6MR+, HOMA values are much larger in 3MR as aromatic and 6MR+ as antiaromatic, whereas HOMA average. These HOMA results suggest that 3MR systems are yields a completely opposite picture. Results from this Table 1 antiaromatic but 6MR+ derivatives are more likely to be aromatic. showcase that different indexes yield different conclusions with Note however that this conclusion drawn from HOMA is contra- regard to which system is aromatic and which is antiaromatic dictory with that from ASE. Using FLU and NICS indexes, we and do not agree with one another. We have observed similar reach different conclusions. For instance, almost all NICS(1) situations in other systems. These results highlight the complexity values are negative in Table 1, indicating that both 3MR and of using different indexes to characterize the aromaticity 6MR+ species are usually aromatic in nature. The same is true properties. This situation can explain in part why there have with the FLU index, where small FLU values are observed in the been confusions in the literature about the concept of aromaticity Table, indicating that both kinds of species are aromatic in as well as how to characterize it.8,9 nature. Put together, these results indicate that (i) substituted Despite the aforementioned perplexity, nevertheless, we fulvene derivatives can be both aromatic and antiaromatic, but observed that within each series, these aromaticity indexes (ii) different aromaticity indexes tend to yield different conclu- are strongly correlated with each other. Shown in Table 2 are sions on which one is aromatic and which one is antiaromatic. correlation coefficient (R) matrices for a few illustrative examples Using data shown in Table 1 as an illustration, FLU and NICS(1) to demonstrate these correlations. A positive value of the correlation indexes characterize both 3MR and 6MR+ as aromatic systems, coefficient R in Table 2 indicates that the two quantities are but HOMA and ASE indicate that 3MR and 6MR+ belong to positively correlated, whereas a negative value points to a negative different camps with regard to aromaticity, ASE categorizes correlation. As can be seen from the Table, FLU and NCIS(1) are Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. Table 1 Numerical results of four aromaticity indexes, FLU, HOMA, NICS, and ASE, for 3MR and 6MR+ systems studied in this work (Scheme 1)

3MR 6MR+ R FLU HOMA NICS(1) ASE FLU HOMA NICS(1) ASE

B(OH)2 0.0267 0.3026 10.1651 0.45 0.0130 0.6677 7.2059 73.28 CC 0.0366 0.1782 6.2511 91.80 0.0029 0.9341 10.7292 130.92 CCH 0.0262 0.2847 10.0413 9.17 0.0099 0.7789 8.2727 84.83 CF3 0.0268 0.2984 10.5419 4.83 0.0159 0.5998 6.4936 61.41 CH3 0.0330 0.0058 8.5331 27.22 0.0100 0.7599 8.4492 89.56 CMe3 0.0344 0.0267 8.3663 24.50 0.0092 0.7672 8.8467 95.51 CN 0.0233 0.3853 10.8816 6.53 0.0144 0.6753 6.7741 62.80 COCH3 0.0224 0.4528 11.1440 14.63 0.0140 0.6581 6.9057 64.45 CONH2 0.0238 0.4080 10.9696 4.99 0.0110 0.7566 8.0119 65.60 COO 0.0349 0.0084 8.1196 50.36 0.0017 0.9640 11.3340 140.35 F 0.0335 0.0328 8.6413 28.64 0.0110 0.7594 8.2250 82.49 H 0.0303 0.0889 8.9791 15.77 0.0131 0.6683 7.4293 68.99 NH 0.0312 0.0987 8.8810 55.05 0.0016 0.9624 11.1881 144.14 + NH3 0.0236 0.3123 11.4243 27.08 0.0230 0.3966 4.3789 16.76 NN+ 0.0317 0.0933 8.8051 52.25 0.0291 0.2235 1.1655 32.12 NO 0.0089 0.7596 13.6375 77.40 0.0059 0.8759 8.9429 77.46 NO2 0.0206 0.5017 9.9818 21.23 0.0154 0.6690 6.5121 61.91 O 0.0214 0.4750 11.4823 19.85 0.0017 0.9625 11.2225 144.88 OCH3 0.0337 0.0206 8.0082 42.52 0.0061 0.8708 9.7681 108.62 OH 0.0370 0.2493 7.4698 48.15 0.0075 0.8399 9.2751 102.24 SiMe3 0.0307 0.1225 9.4052 10.54 0.0107 0.7320 8.1850 84.60

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Table 2 Correlation coeﬃcient (R) matrix between every pair of aromaticity indexes for 3MR, 5MR, 6MR+ and 6MR-substituted fulvene derivative systems. A negative R value indicates that the correlation is conversely proportional to each other, whereas a position R value means that the two quantities are positively correlated

FLU HOMA NICS(1) ASE FLU HOMA NICS(1) ASE 3MR 6MR FLU 1 1 HOMA 0.9762 1 0.9870 1 NICS(1) 0.9478 0.9463 1 0.9378 0.9727 1 ASE 0.9230 0.9014 0.9463 1 0.9866 0.9769 0.9228 1

5MR 6MR+ FLU 1 1 HOMA 0.9904 1 0.9936 1 NICS(1) 0.9178 0.9446 1 0.9935 0.9882 1 ASE 0.9737 0.9763 0.9678 1 0.9768 0.9660 0.9862 1

always positively correlated. So are HOMA and ASE. These trends are these quantities. Earlier, we have demonstrated that these IT true across all series of fulvene derivatives studied in this work. quantities are strongly inter-correlated.45,46,52,58,68 We first Table 2 only shows four of them. As an example, shown in Fig. 1 are examine the correlations among these IT quantities. Table 4 the two strong positive correlations, one between FLU and NCIS(1) displays the correlation coeﬃcient (R) matrix between every and the other between HOMA and ASE, for the 6MR+ series. Note pair of the eight IT quantities for 3MR, 5MR, 6MR+ and that, as shown in Table 2, even though the correlation between each 6MR substituted fulvene derivative systems. As can be seen pair of the four aromaticity indexes is the same across all fulvene from the table, (i) strong correlations between IT quantities are series, their slopes and intercepts are usually vastly diﬀerent,37,39,40 again observed; (ii) these correlations are in good agreement making it impossible to put all data points in one plot and find any with what we discovered earlier for other systems;60,61 and (iii) significant correlation for any of the index pairs. these strong correlations are true for all systems studied in this

Now, let us take a look at the quantities from the work. For example, SS and IF are always negatively correlated r r information-theoretic approach in DFRT for these systems. with each other, with R o 0, but SGBP, IG, R2, and R3 are always Table 3 exhibits the average value of ring carbon atoms for positively correlated, with R 4 0. These different but strongly eight IT quantities in the 5MR substituted fulvene derivatives. correlated simple density functionals provide sensitive measure- As can be seen clearly from the data in the table, with the ments about the electron density distribution of the system and change of the substituent, each of these quantities is slightly thus bestow useful insights into the nature and origin of various fluctuating around its characteristic value. It is this fluctuation physicochemical phenomena including aromaticity and anti- pattern that will reflect the nature of aromaticity and antiaromaticity. As an illustration, Fig. 2 exhibits six strong linear aromaticity of the systems studied. This is because these IT correlations of Shannon entropy with other IT quantities for the quantities are measures of the different nature of the electron 6MR+ series of substituted fulvene derivatives. As can be seen density distribution, so changes due to the aromaticity or the from the figure, S is positively correlated with S , I , and Rr Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. S GBP G 2 2 antiaromaticity process should be reflected in the changes in with R = 0.99, but negatively correlated with IF, E2, and E3 with R2 Z 0.9. One key feature is that not only these strong relation- ships agree well with those uncovered for other systems,60,61 but they are also true across different series studied in this work, as shown in Table 4. Note again that in different series, the slope and intercept of these correlations are often vastly different, making it impossible to put all data points in one single line.39,69 Next, we examine the cross correlation between aromaticity indexes and IT quantities. As the main result of this work, it is believed that changing the patterns of cross correlations between these two sets of the physicochemical and electronic properties, aromaticity indexes and IT quantities, could provide insights into the trends and characteristics of these species, which we know could be both aromatic and antiaromatic. As we have demonstrated in Tables 2 and 4, strong correlations among aromaticity indexes themselves as well as those among IT quantities are applicable to all series of fulvene derivatives,

Fig. 1 Illustrative examples of strong linear correlations between aroma- but it is no longer true for cross correlations, as we will show ticity indexes for 6MR+ systems: (a) between FLU and NICS(1) indexes and below. Table 5 includes the first pattern of the cross correlations (b) between HOMA and ASE indexes. for 3MR, 4MR+, 6MR, and 7MR systems. In this case, for all

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Table 3 The average numerical results of ring carbon atoms for eight information-theoretic quantities in the 5MR substituted fulvene derivative system

r r R SS IF SGBP IG E2 E3 R2 R3

B(OH)2 4.8664 245.7139 39.1661 0.0931 31.5223 1121.3373 6.1882 6.4902 CC 5.0586 245.4606 39.5049 0.1422 31.5067 1120.1708 6.2926 6.7418 CCH 4.8614 245.6830 39.1724 0.0935 31.5209 1121.2026 6.1890 6.4880 CF3 4.8347 245.7449 39.1302 0.0881 31.5224 1121.2645 6.1776 6.4738 CH2 5.2484 245.2901 39.8062 0.1926 31.4898 1119.1097 6.4148 7.8248 CH3 4.8995 245.6163 39.2511 0.1060 31.5186 1120.9040 6.2145 6.5270 CMe3 4.8980 245.5997 39.2515 0.1068 31.5177 1120.8679 6.2162 6.5290 CN 4.8176 245.7468 39.0919 0.0828 31.5230 1121.3529 6.1666 6.4575 COCH3 4.8389 245.7676 39.1242 0.0867 31.5240 1121.4553 6.1751 6.4718 CONH2 4.8372 245.7595 39.1250 0.0870 31.5242 1121.4386 6.1757 6.4729 COO 4.9813 245.5561 39.3775 0.1234 31.5133 1120.5880 6.2519 6.6132 F 4.8773 245.6484 39.2303 0.1023 31.5196 1120.8334 6.2065 6.5137 H 4.8868 245.6839 39.2093 0.0994 31.5203 1121.0501 6.2010 6.5078 NH 5.2217 245.3030 39.7774 0.1876 31.4929 1119.1951 6.3982 7.4505 NH2 4.9665 245.4767 39.3921 0.1268 31.5101 1120.1464 6.2571 6.5952 + NH3 4.7507 245.8029 38.9894 0.0720 31.5250 1121.4623 6.1432 6.4302 NMe2 4.9786 245.4452 39.4109 0.1304 31.5083 1120.1167 6.2646 6.6082 NN+ 4.6704 246.0236 38.8010 0.0454 31.5301 1122.2508 6.0884 6.3616 NO 4.8230 245.7822 39.0967 0.0829 31.5243 1121.4282 6.1670 6.4596 NO2 4.8011 245.8032 39.0668 0.0789 31.5253 1121.5445 6.1587 6.4492 O 5.1969 245.3582 39.7400 0.1805 31.4986 1119.5439 6.3805 7.2579 OCH3 4.9252 245.5729 39.3155 0.1145 31.5157 1120.5287 6.2319 6.5534 OH 4.9215 245.5313 39.3200 0.1168 31.5147 1120.4601 6.2360 6.5567 SiMe3 4.8749 245.6566 39.1956 0.0984 31.5200 1121.1407 6.1989 6.5029

the four fulvene systems studied in this work, SS is positively are usually strong except for a few cases in 4MR, whose correlated with FLU and NICS(1) but it is negatively correlated optimized structures were not ideally planar in those cases. r with HOMA and ASE. This same pattern is true for SGBP, IG, R2, To make our point clearer, shown in Fig. 3 and 4 are four r and R3 as well. For IF, E2, and E3, their changing patterns are representative linear correlations of Shannon entropy and opposite, where they are negatively correlated with FLU and Fisher information for 3MR and 5MR systems with FLU ad NICS(1) but positively correlated with HOMA and ASE. HOMA indexes, respectively. In Fig. 3, as we can see, for 3MR Since aromaticity involves the process of electron density and 5MR, FLU has directly inverse correlations with Shannon 4,5 delocalization, it is comprehensible that Shannon entropy SS, entropy (Fig. 3a and c). The same is true for Fisher information which is a measure of the electron delocalization,52 is positively (Fig. 3b and d). For the HOMA aromaticity index, as shown in correlated with FLU and NICS.6,8,26 This is also true for FLU and Fig. 4, 3MR and 5MR possess opposing correlations as well for NICS to negatively correlate with Fisher information, which is a both Shannon entropy (Fig. 4a and c) and Fisher information gauge of the electron localization.42 On the other hand, as we (Fig. 4b and d). Again, NICS(1) behaves the same way as FLU,26 Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. have shown earlier,42 the first-order approximation of the and ASE does the same as HOMA.39 To show how these correlations information gain is simply the Hirshfeld charge.67 A positive will behave if put in the same plot, Fig. 5 is an example, where we

correlation of aromaticity with IG agrees well with the experimental plot the correlation between ASE and SGBP for 3MR, 5MR, 6MR+, finding that electrophilic aromatic substitutions are more favor- and 7MR series. One can see that from the slope sign perspective able with larger Hirshfeld charge, and thus the information gain.34 they belong to two categories, but their values of the slope are The relative Renyi entropy behaves similarly to the information significantly different, confirming the results discussed above. gain, which is also called the relative Shannon entropy,33,34 so it is So, what does this mean? What do these two drastically r r no surprise that R2 and R3 also positively correlated with FLU and different correlation patterns tell us about the nature of these NICS, the same as IG.ForHOMAandASEindexes,wecanmake fulvene derivatives? These seven substituted fulvene derivatives use of the results unveiled in Table 2 to appreciate the strong cross in Scheme 1 appear to have the same sequence, but their correlations in Table 5. changing patterns of aromaticity indexes with regard to The above pattern, however, is not general and it is not information-theoretic quantities suggest that they should belong applicable to other three fulvene derivatives. Shown in Table 6 to two different categories. These seven planar systems are is their own correlation pattern for the other three fulvene structurally similar with their only difference in the size of the systems, 4MR, 5MR, and 6MR+, which is completely opposite member ring and thus the total number of p electrons involved in 37,39,40 to what has been shown in Table 5. For these systems, SS, SGBP, the conjugation interaction. In the first category, excluding r r IG, R2, and R3 are negatively correlated with FLU and NICS(1) the double bond connected to the substituting group, we find

but positively correlated with HOMA and ASE, whereas IF, E2, that 3MR, 4MR+, 6MR, and 7MR species have a total of 2, 2, 6,

and E3 are positively correlated with FLU and NICS(1) but and 6 p electrons within the ring. These numbers are consistent negatively correlated with HOMA and ASE. These correlations with Hu¨ckel’s 4n + 2 rule of aromaticity.27,70,71 On the other hand,

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Table 4 Correlation coeﬃcient (R) matrix between every pair of information-theoretic quantities for 3MR, 5MR, 6MR+ and 6MR substituted fulvene derivative systems. A negative R values indicates that the correlation is conversely proportional to each other, whereas a position R value means that the two quantities are positively correlated

r r SS IF SGBP IG E2 E3 R2 R3 3MR SS 1 IF 0.9000 1 SGBP 0.9943 0.9333 1 IG 0.9893 0.9483 0.9977 1 E2 0.8951 0.9391 0.9101 0.9283 1 E3 0.8218 0.9374 0.8621 0.8830 0.9645 1 r R2 0.9919 0.9432 0.9983 0.9997 0.9257 0.8761 1 r R3 0.9370 0.8111 0.9173 0.9162 0.8760 0.7695 0.9242 1

5MR SS 1 IF 0.9546 1 SGBP 0.9960 0.9754 1 IG 0.9967 0.9700 0.9985 1 E2 0.9821 0.9533 0.9804 0.9880 1 E3 0.9688 0.9862 0.9831 0.9842 0.9816 1 r R2 0.9974 0.9612 0.9963 0.9992 0.9896 0.9797 1 r R3 0.9120 0.7947 0.8846 0.9034 0.9252 0.8578 0.9191 1

6MR+ SS 1 IF 0.9582 1 SGBP 0.9948 0.9804 1 IG 0.9945 0.9801 0.9998 1 E2 0.8715 0.8473 0.8584 0.8606 1 E3 0.9468 0.9844 0.9650 0.9654 0.9129 1 r R2 0.9943 0.9806 0.9998 1.0000 0.8580 0.9649 1 r R3 0.9955 0.9742 0.9991 0.9995 0.8543 0.9580 0.9994 1

6MR SS 1 IF 0.8527 1 SGBP 0.9973 0.8790 1 IG 0.9948 0.8851 0.9963 1 E2 0.9867 0.9100 0.9903 0.9902 1 E3 0.9418 0.9444 0.9583 0.9619 0.9765 1 r R2 0.9042 0.7824 0.8937 0.9267 0.8942 0.8686 1 r R3 0.7109 0.6100 0.6942 0.7488 0.6999 0.6865 0.9416 1 Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. substituting group) is 4, in good agreement with the 4n rule of antiaromaticity. Based on this observation, in consensus with Hu¨ckel’s 4n + 2 rule of aromaticity and 4n rule of antiaromaticity, we conclude that the systems of 3MR, 4MR+, 6MR, and 7MR are aromatic, whereas 4MR,5MR, and 6MR+ are antiaromatic. This conclusion, however, is in apparent contradiction with the results from the aromaticity indexes such as FLU, HOMA, NICS(1), and ASE.1–3,24,26,41 As we have seen in Table 1 as well as from previous results in the literature,39 using these indexes, we found that depending on the nature of substitution groups, the same ring structure can be attributed as either aromatic or antiaromatic.24,25,49 For example, using ASE as the aromaticity

criterion, for 3MR, when R = CH3, it is aromatic (ASE o 0), but Fig. 2 Illustrative examples of strong linear correlations between Shannon when R is changed to mono nitrogen oxide (NO), it becomes entropy and other information-theoretic quantities for 6MR+ systems. antiaromatic (ASE 4 0) (Table 1). Moreover, using different aromaticity indexes, one can come to different conclusions. For instance, still using the same example above, if we employ

in the second category, the total number of p electrons in 4MR, NICS(1) as the criterion of aromaticity, the species with R = CH3 5MR, and 6MR+ (excluding the double bond connected to the and R = NO substituting groups are both aromatic (NICS o 0).

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Table 5 Correlation coeﬃcient (R) matrix between every pair of information-theoretic quantities and aromaticity indexes for 3MR, 4MR+, 6MR, and 7MR substituted fulvene derivative systems, where the same pattern of correlations is observed

r r SS IF SGBP IG E2 E3 R2 R3 3MR FLU 0.9580 0.9783 0.9786 0.9864 0.9369 0.9150 0.9834 0.8637 HOMA 0.9265 0.9675 0.9465 0.9643 0.9801 0.9556 0.9602 0.8704 NICS(1) 0.9697 0.9146 0.9716 0.9741 0.9279 0.8717 0.9753 0.9312 ASE 0.9637 0.8734 0.9640 0.9600 0.8730 0.8187 0.9630 0.9336

4MR+ FLU 0.9877 0.9536 0.9830 0.9800 0.9616 0.9438 0.9811 0.9780 HOMA 0.9660 0.9744 0.9734 0.9774 0.9839 0.9842 0.9767 0.9683 NICS(1) 0.9312 0.9700 0.9471 0.9443 0.9525 0.9732 0.9441 0.9152 ASE 0.9751 0.9768 0.9854 0.9879 0.9762 0.9889 0.9871 0.9753

6MR FLU 0.9763 0.9110 0.9843 0.9793 0.9889 0.9717 0.8512 0.6314 HOMA 0.9867 0.8888 0.9870 0.9922 0.9903 0.9604 0.9168 0.7346 NICS(1) 0.9596 0.8714 0.9578 0.9780 0.9626 0.9473 0.9769 0.8559 ASE 0.9781 0.8764 0.9828 0.9748 0.9767 0.9431 0.8429 0.6185

7MR FLU 0.5823 0.7314 0.6271 0.6072 0.5115 0.6007 0.5893 0.2798 HOMA 0.8033 0.8933 0.8327 0.8208 0.7499 0.8042 0.8082 0.5607 NICS(1) 0.9697 0.9181 0.9619 0.9702 0.9862 0.9599 0.9753 0.9779 ASE 0.9620 0.9804 0.9737 0.9731 0.9449 0.9586 0.9699 0.8513

Table 6 Correlation coeﬃcient (R) matrix between every pair of information-theoretic quantities and aromaticity indexes for 4MR, 5MR, and 6MR+ substituted fulvene derivative systems, where the same pattern of correlations is observed. Their patterns are completely inverse to those in Table 5

r r SS IF SGBP IG E2 E3 R2 R2 4MR FLU 0.8123 0.5877 0.7928 0.8099 0.7923 0.7245 0.8152 0.7993 HOMA 0.6286 0.3077 0.6504 0.6107 0.3712 0.1531 0.4486 0.1902 NICS(1) 0.8071 0.8232 0.8505 0.8151 0.8119 0.8396 0.7016 0.6460 ASE 0.7654 0.8801 0.8241 0.7842 0.8071 0.8822 0.6587 0.6013

5MR FLU 0.9743 0.9481 0.9749 0.9817 0.9913 0.9776 0.9812 0.8964 HOMA 0.9643 0.9675 0.9745 0.9755 0.9767 0.9849 0.9710 0.8465 NICS(1) 0.9325 0.9909 0.9584 0.9500 0.9219 0.9699 0.9387 0.7468 ASE 0.9760 0.9844 0.9874 0.9886 0.9808 0.9939 0.9854 0.8727 Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. 6MR+ FLU 0.9323 0.9920 0.9624 0.9612 0.8009 0.9714 0.9621 0.9537 HOMA 0.9001 0.9788 0.9373 0.9349 0.7607 0.9555 0.9363 0.9261 NICS(1) 0.9194 0.9874 0.9536 0.9534 0.7675 0.9584 0.9545 0.9468 ASE 0.9421 0.9820 0.9681 0.9686 0.7693 0.9487 0.9694 0.9656

Different conclusions can thus be made with different aromaticity is known that within an electronic system, there are many criteria. competing sources of interactions, such as electron–electron, Aromaticity and antiaromaticity are the extra stability and nuclear–electron, exchange–correlation, etc.,34,60,61 and it is the instability of a planar cyclic structure due to the additional compromising effort of all these interactions working together delocalization of electrons, leading to the re-distribution of the that yield a stationary state of the system. Using one index electron density.45,52,53 It is consistent with the discussion by to gauge the overall stability of a system can, at best, only Frenking et al.27 and elaboration a bit further by Grande-Aztatzi measure one aspect of the cooperative effort of the entire system. et al.73 that not only the number of delocalized electrons needs Each aromaticity index reflects the characteristics of one aspect. to be assessed, but also the nature of the occupied orbitals is as They are not necessarily always in agreement to one another. important and needs to be paid close inspection in order to This work provides with us a novel view to this complicated claim aromaticity/antiaromaticity. The characteristics and phenomenon from the perspective of information theory. It is peculiar properties of aromatic systems as reflected by diﬀerent our humble hope that this work adds up to the continuing aromaticity indexes such as FLU, HOMO, NICS and ASE are efforts in the literature and ultimately facilitates our search for a some of the consequence, not the cause, of aromaticity. It comprehensive and integrated view of aromaticity.

PCCP Paper 5. Concluding remarks

From the context of its original definition, aromaticity characterizes the extra stability of a cyclic planar structure with a delocalized conjugated system of 4n +2p electrons. As more kinds of aromaticity such as polycyclic aromaticity, Mo¨bius aromaticity, metalloaromaticity, etc., were unveiled and more featured properties through aromaticity descriptors in structure, energy and magnetism were discovered, the definition lost its original meaning. Though these aromaticity indexes are able to describe some properties, given the broad range of diversified systems involved as well as many different kinds of properties required to be taken into account, it is understandable that none of these descriptors is able to adequately take care of all of them. This stabilization effect is only one of the many competing effects in Fig. 3 Illustrative examples of strong linear correlations between the aromaticity index FLU and information-theoretic quantities such as an electronic system. To single out a descriptor for its adequate Shannon entropy and Fisher information for different substituted fulvene quantification is not uncomplicated, if ever possible. derivatives. In this work, we examined the issue in a four-dimensional manner. We considered (i) a series of fulvene derivatives (3MR– 7MR), with (ii) 24 possible substitution groups, and investigated the changing patterns of both (iii) four representative aromaticity indexes (ASE, FLU, HOMA, ASE and NICS) and (iv) eight information-theoretic quantities. This way of looking into the matter provided with us information unavailable before in the literature. We found that aromaticity indexes are often well correlated with one another. The same is true for IT quantities. Plus, these correlations are valid across all series of fulvene derivatives with different ring structures. For cross correlations between IT quantities and aromaticity indexes, however, we showed that, even though the correlations were usually strong, two completely opposite correlations were observed, meaning that these correlations are not valid across all series of fulvene derivatives. The nature of these correlations depends on the nature of the ring structure. The two groups of the systems each obeying the same cross-correlation patterns happen to have a Fig. 4 Illustrative examples of strong linear correlations between the total number of 4n + 2 and 4n p electrons, respectively, which Published on 22 June 2017. Downloaded by UNIVERSITEITSBIBIOTHEEK VUB 28/12/2017 09:06:26. aromaticity index HOMA and information-theoretic quantities such as are in consensus with Hu¨ckel’s rule of aromaticity and anti- Shannon entropy and Fisher information for different substituted fulvene aromaticity. These results confirm that there is no one aromaticity derivatives. indicator and that perhaps it should be considered aromaticity as a multidimensional concept.72 These results also suggest that information-theoretic quantities can provide extra insight not available from other studies about the nature of aromaticity and antiaromaticity. Whether or not these patterns are still valid in other aromatic systems and what their general implications could be are currently still under investigation, whose results will be published elsewhere.

Acknowledgements

CYR and SBL acknowledge support from the National Natural Science Foundation of China (No. 21503076) and Hunan Provincial Natural Science Foundation of China (Grant No.

Fig. 5 Illustrative examples of strong linear correlations between the 2017JJ3201). DHY acknowledges the support from the Hunan

aromaticity index ASE and the information-theoretic quantities SGBP for Provincial Innovation Foundation for Postgraduates (No. substituted fulvene derivatives 3MR, 5MR, 6MR+, and 7MR. CX2017B179).

Paper PCCP

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