A universal scale of aromaticity for π-organic compounds

A Universal Scale of Aromaticity for

p-Organic Compounds

MERCEDES ALONSO, BERNARDO HERRADON

Instituto de Quımica Organica General, CSIC, Juan de la Cierva 3, 28006 Madrid, Spain

Received 8 May 2009; Revised 10 June 2009; Accepted 10 June 2009DOI 10.1002/jcc.21377

Published online 27 July 2009 in Wiley InterScience (www.interscience.wiley.com).

Abstract: Aromaticity is an essential concept in chemistry, invented to account for the stability, reactivity, molec-

ular structure, and properties of many organic and inorganic compounds. In recent years, numerous methods to quan-

tify aromaticity based on the energetic, magnetic, structural, and electronic properties of molecules have been pro-

posed but none of them is universal. The inability of establishing a universal scale of aromaticity based on a single

parameter is due to the multidimensional character of this phenomenon. Consequently, aromaticity analyses should

be carried out by employing a set of aromaticity descriptors on the basis of different physical manifestations of aro-

maticity. Here, we report a universal scale of aromaticity for p-organic compounds based on the Euclidean distance

between neurons in a self-organizing map. The most widely used aromaticity indicators have been used as molecular

descriptors, and so our approach provides the first scale of aromaticity which contains the energetic, magnetic, and

structural aspects of this property. The method is applicable to a wide variety of unsaturated organic compounds and

allows quantification of both aromaticity and antiaromaticity. Additionally, the position of a compound on the bi-

dimensional map determinates immediately the following: (a) the group (aromatic, nonaromatic, or antiaromatic) to

which the system belongs, (b) their degree of p-electronic delocalization, and (c) the similarity in aromaticity/anti-

aromaticity between different compounds. This new scale of aromaticity is able to indicate the expected order of

aromaticity of analogues of fulvene and heptafulvene, heteroaromatic species, substituted benzenes, and functional-

ized cyclopentadienyl compounds.

q 2009 Wiley Periodicals, Inc. J Comput Chem 31: 917–928, 2010

Key words: aromaticity; neural network; aromaticity indices; antiaromaticity; NICS scan

Introduction

Aromaticity is one of the most important concepts in chemis-

try.1–5 Although it was first studied with organic molecules, it

has been recently applied to inorganic compounds.6,7 Similar to

other fundamental chemical properties such as charge, electrone-

gativity, or bond order, aromaticity is not an observable one and

it has to be defined by convention and measured by a relative

scale.8,9

Although aromaticity is a unique physical phenomenon, it is

manifested in multiple ways. Some consequences of aromaticity

are energetic stabilization as well as particular chemical reactiv-

ity, geometry, magnetic and electronic properties.10,11 Its differ-

ent manifestations have lead to multiple ways for assessing the

aromatic or antiaromatic character of molecules.12 Thus, several

aromaticity indices have been proposed in the literature, which

give distinct orders of aromaticity.13,14 Accordingly, most

authors advise to use a set of aromaticity descriptors based on

different physical properties to characterize aromatic com-

pounds.1,14,15 This fact has prompted a lively discussion on the

monodimensional or multidimensional character of aromatic-

ity.16–19 As pointed out earlier, we consider that aromaticity is a

single physical phenomenon with multiple manifestations, that

is, a multifaceted property, which might be measured by a single

scale that takes into account the different dimensions.

Why aromaticity has to be measured? First, by a pure scien-

tific reason because each property has to be quantified. From a

more practical point of view, arenes are key components in bio-

logically active compounds and technological useful materials

which have deeply influenced on human welfare.20 The most

fundamental property of arenes is aromaticity which is widely

used for understanding the molecular structure, chemical bond-

Additional Supporting Information may be found in the online version of

this article.

Correspondence to: M. Alonso or B. Herradon; e-mail: mercuea@

iqog.csic.es or [email protected]

Contract/grant sponsor: The Spanish Ministry of Education and Science;

contract/grant number: CTQ2007-64891/BQU

Contract/grant sponsor: Spanish Ministry of Education and Science

(MEC) (FPU fellowship)

q 2009 Wiley Periodicals, Inc.

ing, and properties as well as to predict stability and reactivity

of numerous organic and inorganic compounds. Therefore, a

quantitative scale of aromaticity is highly valuable for the design

of new molecules and materials.1–3,8 This is the goal of this

research.

Aromaticity has been measured using a variety of indices

based on geometrical,11,21 energetic,22,23 electronic,24–26 and

magnetic27,28 criteria. However, no fully satisfactory result has

yet been achieved because of its multiple manifestations, con-

cluding that there is not yet a single indicator of aromaticity that

works properly for all cases.9 Although, in principle, we can

consider that aromaticity might be expressed by a lineal combi-

nation of an arbitrary number of indices, this approach has met

with limited success13,14,29 which likely indicates that the rela-

tionship between aromaticity and the different criteria (or mani-

festations of the phenomenon) is nonlinear (see Supporting In-

formation).

Neural networks are useful mathematical tools for modeling

nonlinear functions in many real-world applications, without

knowing the analytic forms in advance.30 These artificial sys-

tems emulate the function of the brain, where a very high num-

ber of information-processing neurons are interconnected and

are known for their efficiency in self-organization, pattern recog-

nition, and dimensionality reduction. They are capable of learn-

ing complex interactions among the input variables even when

they are difficult to find and describe. Consequently, neural net-

works can provide solutions for problems that do not have an

algorithmic solution. They have been intensively used in differ-

ent fields of knowledge such as biology, engineering, and econ-

omy. In chemistry, the use of neural networks has further

expanded into the analysis of spectral data, drug design, predic-

tion of chemical reactivity and physical properties as well as the

development of quantitative structure–activity relationship.31–36

Some advantages of neural networks, useful for the purpose

of this work, are as follows: (i) they are adaptative, i.e., they

can take data and learn from it; (ii) they are essentially nonlin-

ear; (iii) they are capable of generalization, i.e., they can cor-

rectly process information that only broadly resembles the origi-

nal data training; (iv) they are fault-tolerant being capable of

properly handling noisy or incomplete data.

Recently, we reported the successful applications of Koho-

nen’s self-organizing maps (SOMs), an unsupervised neural net-

work, to classify five-membered heterocycles according to their

aromatic/antiaromatic character.37 Some useful characteristics of

the proposed method for measuring aromaticity are as follows:

(1) it is very fast being practically instantaneous to place a new

compound in the map; (2) the placement of the different com-

pounds is conveniently visualized; (3) it has predictive power.

In this article, we expand this methodology to a full set of

organic compounds ranging from highly aromatic to highly anti-

aromatic (Chart 1). The method is applicable to molecules of

different sizes (five-, six-, seven-, and eight-membered rings),

both carbocyclic and heterocyclic, including neutral, cationic, or

anionic species, and it allows us to quantify both aromaticity

and antiaromaticity. Since the input data in the neural network

are diverse energetic, magnetic, and structural descriptors of aro-

maticity, the SOM represents the multidimensional character of

aromaticity and, more importantly, the Euclidean distance

between neurons is a universal scale of aromaticity for p-organiccompounds which takes into account the different physical mani-

festations of this phenomenon.

Computational Details

All calculations have been performed with Gaussian03 package

of program at the MP2 level of theory with the 6-3111G(d,p)

basis set.38 The geometries of five- and six-membered rings,

penta- and heptafulvene and analogues, substituted benzenes,

and substituted cyclopentadienyl compounds were fully opti-

mized and characterized by harmonic vibrational frequency com-

putation, which showed that all structures were minima on the

potential energy surface. Four well strain-balanced homodes-

motic reactions based on cyclic reference compounds were used

to estimate the aromatic stabilization energy (ASE) of five- and

six-membered rings (Scheme 1).39,40 The energies were cor-

rected by MP2/6-3111G(d,p) zero-point energies. Systems with

largely enhanced aromatic stabilization energies (positive values

of ASE) are aromatic, whereas those with strongly negative

ASE values are considered to be antiaromatic.22

For the evaluation of aromatic stabilization energies of hepta-

fulvene and analogues, cycloheptatrienyl cation, and cycloocta-

tetraenyl dication, we have used the isomerization method (ISE),

which is based on the energy differences between a methyl de-

rivative of the conjugated system and its nonconjugated exo-

cyclic methylene isomer (Scheme 2).41 As one closely related

reference compound is involved, ISE provides excellent esti-

mates of aromatic stabilization energies that minimize perturbing

influences such as strain.

The magnetic susceptibility exaltation (L) is defined as the

difference between the magnetic susceptibility of a compound

(vM) and a reference one without cyclic electron delocalization

(vM0) [eq. (1)].42,43 The exaltations were obtained from the reac-

tions indicated in Schemes 1 and 2. The magnetic suscepti-

bilities were computed using the CSGT method44 at the HF/

6-3111G** level of theory. The exaltations are negative (dia-

magnetic) for aromatic compounds and positive (paramagnetic)

for antiaromatic compounds.

K ¼ vM � vM0 (1)

As additional magnetic indices of aromaticity, we have used

three derivations of the nucleus-independent chemical shift

(NICS).45 This index is defined as the negative value of the

absolute magnetic shielding computed at ring centers or another

interesting point of the system.46 NICS values have been com-

puted at the ring centers determined by the nonweighted mean

of the heavy atoms coordinates. The NICS(1) values calculated

1 A above the molecular plane are considered to better reflect

the p-effects. Another descriptor is the out-of-plane component

of the tensor NICS computed at 1 A above the ring center,

denoted as NICSzz(1), which was recently found to be a good

measure for the characterization of the p system of the ring. The

GIAO/HF/6-3111G(d,p) method were used for the NICS calcu-

lations.47 Rings with highly negative values of NICS are quanti-

918 Alonso and Herradon • Vol. 31, No. 5 • Journal of Computational Chemistry

Journal of Computational Chemistry DOI 10.1002/jcc

fied as aromatic, whereas those with positive values are antiaro-

matic. For 25 compounds of the training and validation dataset,

the NICS values have been also evaluated in the ring critical

point (rcp),48 as suggested by Morao et al.49 The results at the

rcp are practically identical to those calculated at the geometri-

cal ring center and they are reported in the supporting informa-

tion (Supporting Information Table S1 and Figure S1).

Isotropic NICS values, used as descriptors in the former

classification,37 do not describe correctly the aromatic/anti-

aromatic character for most of the derivatives of cyclo-

pentadienyl anion and cation.50 As previously reported, these

magnetic indices contain large influences from the r system

and from all three principal components of the NICS tensor.51

For the NICS scans, we have plotted the NICS values along

the z-axis to the ring plane beginning on the geometric center

of the molecular ring up to 5.0 A at intervals of 0.1 A. The

eigenvalues of the chemical shift tensors were used to separate

the isotropic NICS values into their in-plane and out-of-plane

components.52,53

As a structure-based measure, we have employed the har-

monic oscillator model of aromaticity (HOMA) defined by Krus-

zewski and Krygowski as follows54,55:

Chart 1. Five- and six-membered ring compounds used in the training of the neural network. Com-

pounds marked with an asterisk were used in the validation of the self-organizing map. The structures

of additional test compounds are indicated in Table 2.

919Scale of Aromaticity for p-Organic Compounds


HOMA ¼ 1� an

Xn

i¼1

ðRopt � RiÞ2 (2)

where n is the number of bonds taken into the summation, and a is

an empirical constant fixed to give HOMA 5 0 for a model nonar-

omatic system and HOMA 5 1 for a system with all bonds equal

to an optimal value Ropt, assumed to be realized for fully aromatic

system. Ri is the running bond length. This index was found to be

one of the most effective structural indicators of aromaticity.11

Once the aromaticity descriptors have been obtained, we

have generated a family of input vectors which represent each

compound of the training dataset.

The Kohonen networks were obtained with the SOM_PAK

program.56 About 50 SOMs were trained varying both the map

size (number of neurons) and training parameters. Two different

topologies, namely rectangular and hexagonal, were also tested

finding that the hexagonal lattice was better for visual inspec-

tion. The quantization error was used to evaluate the perform-

ance of the network, which is the average distance between the

descriptor vectors and their respective map neurons:

Error ¼

PP

p¼1

PN

i¼1

ðxpi � wjiÞ2

P(3)

The best map is expected to yield the smallest average quan-

tization error with respect to the training and validation set. In

this case, a hexagonal lattice with 26 3 16 neurons using Bub-

ble function as neighborhood kernel was selected. Training was

done in two phases: an ordering phase with 2000 steps and a

self-organizing phase with 20,000 steps. During the first cycle, it

carried out 2000 training steps and the learning rate and initial

neighborhood were set to 1 and 10, respectively. The parameter

values for the second cycle were 0.05 and 1, respectively.

Results and Discussion

In this pattern, each compound is represented by four widely

used aromaticity descriptors based on energetic, magnetic, and

structural manifestations of aromaticity: the ASE, the magnetic

susceptibility exaltation (L), the out-of-plane component of the

NICS tensor computed at 1 A above the ring center [NICSzz(1)],

Scheme 2. Isodesmic reaction used to evaluate the isomerization

energies (ISE) and magnetic susceptibility exaltation (L) of hepta-

fulvene and analogues, C7H71 and C8H8

21.

Scheme 1. Homodesmotic reactions used to evaluate the aromatic stabilization energies (ASE) and

magnetic susceptibility exaltation (L) of five-membered heterocyclic compounds (a), substituted cyclo-

pentadienyl compounds (b), six-membered heterocycles (c), and substituted benzenes (d).



and the HOMA. We have previously demonstrated that these

indices are the most suitable for describing the aromaticity/anti-

aromaticity of a wide variety of organic compounds50 and, more

importantly, they are readily calculated and implemented in

many quantum chemistry programs. In the former classification

we used the isotropic values of NICS, but NICSzz(1) was found

to be the best NICS-based indicator of aromaticity for substi-

tuted cyclopentadienyl compounds, because the local r-bondingcontributions are diminished and the effects of the p-electronring current are dominant.

Chart 1 shows the molecular structures of the 150 five- and

six-membered rings used in the training and validation of the

network. This comprehensive dataset contains a wide range of

aromatic, nonaromatic, and antiaromatic compounds. The values

of the ASE, L, NICSzz(1), and HOMA for each system are

given in Supporting Information Table S2. These descriptors are

used as a four-dimensional input vector for the SOM, which

creates a nonlinear projection onto a two-dimensional space pre-

serving topological relations as faithfully as possible. SOM

automatically clusters similar compounds based on these

descriptors. A 21 3 16 output grid with hexagonal boundaries

and the lowest quantization error (0.92) was chosen. The ratio of

the size of the network when compared with the size of the

dataset is bigger than the previously reported one37 in order to

increase the interpolation capability of SOM for new com-

pounds. A visually effective way for showing the results is a

Sammon map (Fig. 1), a nonlinear projection generated through

an iterative procedure.57,58 In this map, the four-dimensional

input vectors are reduced to points with coordinates (x, y), sothe distances among points corresponds to Euclidean distances

between neurons in the SOM. Accordingly, Sammon map allows

the visualization of the shape of the clusters and the relative

distance between them.

The map shows fairly well the quantitative character of the

classification obtained. The neuron activated by benzene (144)

represents the highest degree of aromaticity, whereas the neuron

activated by singlet cyclopentadienyl cation (35) represents the

highest degree of antiaromaticity. Aromaticity decreases gradu-

ally from benzene to cyclopentadienyl cation. Accordingly, the

Euclidean distance to the neuron activated by benzene (dj), thereference aromatic compound, can be used as a quantitative

measurement of aromaticity/antiaromaticity in five- and six-

membered rings. It is important to note that many aromaticity

descriptors have a series of disadvantages when comparing rings

of different size. For example, magnetic susceptibility exaltation,

and to a lesser extent NICS, depends heavily on the ring area59

and ASE is strongly dependent on the reaction scheme which is

necessarily different for five- and six-membered rings.22,40

The U-matrix, the matrix of distances between adjacent units,

was used to analyze the distribution of classes of the trained

SOM (Fig. 2). The distances between the neighboring neurons

are visualized by gray levels, so the color difference indicates

dissimilarities or similarities between weight vectors of neigh-

borhood neurons in the SOM. If color changes between adjacent

neurons are smooth the neurons are more similar than when

color changes are sharp. Therefore, clusters are white zones sur-

rounded by black boundaries. Figure 2 shows three large clusters

on the left-hand side, middle region, and right-hand side.

Compounds placed on the left cluster are stabilized energetically

and exhibit diamagnetic susceptibility exaltation and highly neg-

Figure 1. Sammon map obtained for the training dataset, showing the relative distances between the

input variables [ASE, L, NICSzz(1), and HOMA] in the original space. The color scale indicates the

Euclidean distances between the weight vector of each neuron and the neuron activated by benzene.



ative values of NICSzz(1) as well as practically equalized bond

lengths (HOMA values ranges from 0.6 to 1), that is, aromatic

compounds. Benzene (144), pyridine (145), pyrrole (3), thio-

phene (2), furan (1), and substituted cyclopentadienyl anions

(96–119) are some compounds located in this region. On the

contrary, right cluster includes antiaromatic compounds, which

are destabilized and exhibit paramagnetic susceptibility exalta-

tion and the single and double bond lengths are localized, such

as borole (31), 1H-silolylium (43), and all the substituted deriva-

tives of cyclopentadienyl cation (120–143). Rings located on the

middle side can be classified as nonaromatic and show interme-

diate values for ASE, L, NICSzz(1), and HOMA. Some repre-

sentative compounds belonging to this class are phosphole (4),

arsole (53), cyclohexadiene, fulvene, and all the substituted

cyclopentadienes (96–119).

Consequently, from energetic, magnetic, and structural

descriptors, SOM automatically classifies the extensive dataset

of 150 five- and six-membered rings into three classes: aromatic,

nonaromatic, and antiaromatic. The border between these three

classes is well defined, so the challenging problematic classifica-

tion of borderline compounds is overcome by using the Kohonen

neural network. According to the neural network, the Euclidean

distance ranges from 0 to 33.2 for aromatic systems, from 35.3

to 68.3 for nonaromatic compounds, and from 72.6 to 170.6 for

antiaromatic molecules.

Additionally, SOM places compounds with similar but non-

identical aromatic character in neighboring neurons, creating a

smooth transition of different aromaticity degrees over the whole

map. Figure 3a shows the trained SOM with the neurons color

coded. The color scale indicates the Euclidean distances between

the weight vectors of each neuron and the neuron activated by

benzene (144). White neurons represent the cluster boundaries

determined by the U-matrix map as discussed earlier.

Table 1 collects the Euclidean distance values for compounds

1–150 used in the training and validation of the network. In con-

trast to our former classification (used mostly for five-membered

rings and not trained for highly aromatic and antiaromatic sys-

tems),37 the new SOM is able to quantify the aromaticity for six-

membered ring as well as antiaromaticity for derivatives of cyclo-

pentadienyl cation. According to the new scale of aromaticity,

benzene and pyridine are the most aromatic six-membered rings

(dj 5 0.0), followed by pyrazine (dj 5 10.2), pyridazine (dj 511.4), and finally, pyrimidine (dj 5 16.8). To discern among ben-

zene and pyridine, a larger map is necessary which assigns a dj 54.1 to the latter (see below). In relation to derivatives of cyclopen-

tadienyl cation, the dj values indicate that the substituents signifi-

cantly reduce the antiaromaticity of 35, especially the electron-

donating hydroxyl group, decreasing dj from 170.6 to 78.0. In

fact, SOM places the 2,5- (129) and 2,3,5- (137) hydroxylated

derivatives in the border with nonaromatic compounds.

Figure 2. U-matrix representation of the Kohonen network: the distances between neighboring neurons

are visualized by gray levels. Darker hexagons indicate a large distance.



Figure 3. (a) Kohonen network obtained for the classification of 150 five- and six-membered rings

(1–150) according to their degree of aromaticity/antiaromaticity. The numbers in brackets represent the

neuron coordinates. Neurons are colored based on the Euclidean distance between the weight vectors

of each neuron and the neuron activated by benzene (144) and pyridine (145). White numbers repre-

sent the compounds used in the validation of the network and white neurons represent the cluster

boundaries. (b) Extension of the region of highly aromatic compounds with dj between 0 and 10,

obtained from a 14 3 12 map training only with aromatic compounds.

Figure 4. NICS values as a function of distance of benzene (144), 2,3,4,5-tetrahydroxycyclopenta-

dienyl anion (117), 3-phospaphosphole (12), gallole (47), 2,5-dihydroxycyclopentadienyl cation (129),

and cyclopentadienyl cation (35): (black circles) Isotropic chemical shift; (green circles) Out-of-plane

component; (red circles) In-plane component.

We have carried out additional tests to check the universality

of our method for quantifying aromaticity. Sola and coworkers

have recently established several tests to assess the performance

of the existing indices of aromaticity (Table 2).60 Only the tests

related to the aromaticity of monocycles have been proved.

According to the nature of the aromaticity indices used as

descriptors in the input vectors of the SOM, the Euclidean dis-

tance is a measure of the global aromaticity of the molecule.

Interestingly, the possibility of introducing compounds lacking

some of the descriptors allows to apply the neural network to

analyze also local aromaticity based on the HOMA and

NICSzz(1) values. It would be advisable to generate another

Kohonen network for measuring local aromaticity which takes

into account, apart from the structural and magnetic indicators,

several indices of electronic such as the aromatic fluctuation

index or the multicenter indices. However, the high computa-

tional cost of electronic indices involves a drawback for the

method functionality.61

The first test involves the study of aromaticity in a set of

substituted analogues of fulvene (Table 2A). It is well known

that the p-electron delocalization in analogues of fulvene

depends strongly on the electronegativity of the X substituents.

Electronic effects of these systems are usually rationalized in

terms of their differing weights of the zwitterionic resonance

structures. Electron-donating substituents enhance the aromatic-

ity of pentafulvenes by increasing the weight of the 6p-electronstructure A, whereas electron withdrawing substituents increase

the antiaromaticity of the ring favoring 4p-electron structure B

(Scheme 3).62 Therefore, a good aromaticity measure should

give the following order of aromaticity for the analogues of

fulvene: BH22 [ CH2 [ NH [ O [ NH2

1. This is exactly the

order provided by the Euclidean distance values, ranging from

18.5 to 82.0 (Table 2A). Additionally, SOM places compound

151 (X 5 BH22) in the region of the highly aromatic com-

pounds while 153 (X 5 NH21) is mapped into an antiaromatic

neuron.

Table 1. Euclidean Distance Values (dj) and Coordinates of the Winner Neuron for Compounds 1–150.a,b,c

Aromatic Nonaromatic Antiaromatic

dj Compound Neuron dj Compound Neuron dj Compound Neuron dj Compound Neuron dj Compound Neuron

0.0 104 (0, 0) 18.8 26 (4, 2) 35.9 12 (6, 9) 46.0 70* (11, 12) 72.6 129 (13, 0)

0.0 145 (0, 0) 18.8 27 (4, 2) 36.2 49 (8, 5) 46.3 56 (12, 11) 78.0 137 (14, 2)

9.9 107 (0, 2) 18.9 112 (5, 3) 36.6 4 (8, 6) 46.3 69 (12, 11) 87.6 50 (20, 11)

10.2 146 (0, 3) 19.0 21 (4, 4) 37.4 8 (7, 7) 46.3 73 (12, 11) 89.0 31 (17, 6)

10.3 99 (1, 2) 19.3 15* (3, 6) 38.6 16 (8, 8) 46.4 80 (11, 7) 90.8 43 (20, 9)

11.4 45 (0, 4) 19.3 110 (3, 6) 41.0 53 (10, 4) 46.4 88 (11, 7) 99.6 121 (15, 4)

11.4 52 (0, 4) 19.4 106 (1, 8) 41.5 33 (6, 11) 46.4 92* (11, 7) 101.4 130 (15, 0)

11.4 148* (0, 4) 20.1 101 (2, 7) 41.6 63 (8, 9) 47.9 87 (11, 15) 101.7 141 (20, 7)

12.3 18 (2, 2) 20.3 17 (6, 3) 42.0 64 (8, 10) 49.3 149 (11, 2) 103.3 138 (15, 3)

12.4 24 (2, 1) 20.3 11 (9, 0) 42.1 71 (7, 12) 49.3 150* (11, 2) 113.2 128 (16, 3)

12.7 28 (2, 0) 20.5 105 (0, 9) 42.2 67 (5, 13) 49.5 86 (13, 14) 116.1 125 (18, 7)

13.0 22 (3, 0) 20.5 14 (2, 8) 42.3 76 (9, 10) 49.7 95 (13, 15) 116.1 142* (18, 7)

13.1 34 (0, 5) 20.7 114 (1, 9) 42.7 65 (8, 12) 50.0 82 (13, 13) 122.7 122 (17, 3)

14.2 20 (2, 3) 21.0 23 (4, 5) 42.7 66 (8, 12) 50.5 93 (14, 13) 122.7 136* (17, 3)

14.3 3 (4, 0) 21.0 108* (4, 5) 43.2 38 (6, 15) 50.7 90 (15, 15) 131.1 120 (18, 3)

14.3 96 (4, 0) 21.3 1 (7, 4) 43.2 32 (10, 6) 50.9 48 (11, 5) 132.2 140 (19, 4)

15.1 37 (1, 5) 22.7 19 (6, 5) 43.2 83 (9, 12) 51.2 81 (14, 12) 132.3 132 (18, 5)

15.5 98 (0, 6) 22.9 116 (5, 5) 43.4 60 (9, 9) 52.2 41 (13, 4) 134.4 134 (19, 5)

15.7 103 (5, 0) 23.5 9 (4, 7) 43.5 77 (8, 14) 52.2 51* (13, 4) 137.5 133 (20, 5)

16.3 2 (6, 0) 23.6 5 (5, 8) 44.1 68 (10, 8) 52.4 89 (17, 14) 146.5 124 (20, 3)

16.3 115 (6, 0) 24.2 109 (2, 11) 44.1 72 (10, 8) 53.6 44 (13, 9) 146.5 126 (20, 3)

16.4 6 (5, 1) 24.2 118 (2, 11) 44.5 84 (10, 9) 53.7 94 (18, 15) 149.7 139 (17, 0)

16.4 104 (5, 1) 24.8 111 (0, 11) 45.0 75 (9, 13) 54.0 54 (14, 10) 153.8 131 (18, 0)

16.6 7* (4, 1) 25.3 119 (0, 13) 45.2 61 (11, 10) 54.5 46 (15, 10) 156.1 135 (18, 1)

16.8 147* (7, 0) 25.6 113 (4, 12) 45.2 62 (11, 10) 59.7 39 (13, 7) 160.6 143 (19, 0)

17.2 10 (0, 7) 25.8 42* (1, 14) 45.7 85 (10, 11) 62.9 36 (16, 11) 167.8 123 (20, 0)

17.2 102 (0, 7) 27.5 25 (3, 9) 45.7 78 (9, 15) 65.4 29* (14, 6) 167.8 127 (20, 0)

17.8 97 (2, 6) 29.0 13 (5, 10) 45.7 79 (9, 15) 66.7 58 (16, 9) 170.6 35 (20, 1)

18.1 100 (5, 2) 30.4 30* (0, 15) 45.9 91 (10, 14) 67.6 57 (18, 10)

18.4 55 (7, 1) 33.1 117 (3, 14) 46.0 74 (11, 12) 67.7 59 (15, 8)

68.2 47 (15, 6)

aThe structures of compounds 1–150 are indicated in Chart 1.bThe compounds marked with an asterisk were used in the validation of the neural network. Six-membered rings are

highlighted in gray.c(x, y) represent the neuron coordinates, where x is the column number and y is the row number (Figure 2).



Opposite variation of aromaticity with the electronegativity of

the X substituent is expected for analogues of heptafulvene

because the dipolar structures with 6p electrons are oppositely

polarised for the five- and seven-membered rings.63 Accordingly,

dj values indicate a gradual decrease of aromaticity from the

aromatic 6p tropylium cation C of 158 (X 5 NH21) to the antiaro-

matic 8p anionic ring D of 156 (X 5 BH22), as shown in Table 2B.

A third test analyzes a series of six five-membered hetero-

cycles (Table 2C). Electron counting indicates that systems with

X 5 CH2, NH, O, and S are aromatic and it is expected that

the inclusion of heteroatoms decreases the aromaticity with

regard to the cyclopentadienyl anion in the order NH \ S \

Table 2. Calculated ASE (kcal mol21), L (ppm cgs), NICSzz(1) (ppm), HOMA, and Euclidean Distance (dj)

for Test Compounds.

Compound ASE L NICSzz(1) HOMA dj Compound ASE L NICSzz(1) HOMA dj dja

(A) Analogues of fulvene (D) Six-membered heterocyclesb

X

BH22 14.04 211.10 229.59 0.483 18.5 Pyridine 31.63 212.53 230.86 0.978 0.0 4.1

CH2 23.06 1.01 26.97 20.142 46.3 Pyridazine 22.17 29.91 230.98 0.962 11.4 13.7

NH 29.29 4.97 2.55 20.592 58.8 Pyrimidine 19.39 212.82 228.64 0.991 16.8 15.9

O 214.65 8.00 10.96 21.255 67.6 Pyrazine 23.29 210.19 230.95 1.000 10.2 12.5

NH21 227.27 11.75 25.71 20.662 82.0 s-Triazine 9.63 27.90 225.63 0.995 25.8 25.8

(B) Analogues of heptafulvene (E) Substituted benzenes

X

X

BH22 0.27 34.10 74.07 0.175 123.7 N2

1 31.50 213.39 228.84 0.955 0.0 4.7

CH2 6.78 2.27 14.97 0.335 55.9 CN 33.72 213.89 230.70 0.951 0.0 1.9

NH 9.52 2.30 2.23 0.390 48.2 F 34.16 213.66 231.07 0.974 0.0 1.9

O 12.71 25.65 27.93 0.421 30.3 H 34.83 214.95 231.84 0.962 0.0 0.0

NH21 21.32 210.09 219.12 0.862 20.3 CH3 33.70 213.36 230.67 0.955 0.0 2.3

NH2 34.06 211.20 228.54 0.957 0.0 5.1

OH 33.31 212.88 229.85 0.966 0.0 2.8

(C) Five-membered heterocycles (F) 6p-electron systems

X

X

CH2 22.05 210.13 235.98 0.740 13.1 benceno 36.85 215.35 231.84 0.962 0.0 0.0

NH 20.57 26.48 232.41 0.876 14.3 C7H72 34.17 212.46 229.00 0.954 0.0 3.5

S 18.57 27.00 229.41 0.891 16.3 C8H822 5.85 26.27 227.33 0.808 25.8 23.2

O 14.77 22.90 226.79 0.298 21.3

CH2 0.00 0.00 211.83 20.778 41.5

BH 222.49 16.09 31.30 20.595 89.0

CH1 257.88 35.54 112.21 21.152 170.6

aEuclidean distance values obtained from the larger map, trained only with aromatic compounds.bThe structures of the rest of six-membered heterocycles are shown in Chart 1.

Scheme 3. Resonance structures for analogues of fulvene and hepta-

fulvene.



O.3,64 Cyclopentadiene (X 5 CH2) is anticipated to be a nonaro-

matic compound, whereas compounds with X 5 BH and CH1

should be antiaromatic. The Kohonen network classifies properly

all the five-membered heterocycles and the Euclidean distance

values give the expected order of aromaticity for this set of mol-

ecules: 34 (X 5 CH2) [ 3 (X 5 NH) [ 2 (X 5 S) [ 1 (X 5O) [ 33 (X 5 CH2) [ 31 (X 5 BH) [ 35 (X 5 CH1). By

contrast, the inclusion of nitrogen atoms at position 2 in the pyr-

role ring increases the aromaticity, being 1H-1,2-diazole (18, dj5 12.3), 1H-1,2,3-triazole (24, dj 5 12.4), and 2H-1,2,3,4-tetra-zole (28, dj 5 12.7) more aromatic than the cyclopentadienyl

anion (34, dj 5 13.1).

For six-membered heterocycles, it is expected that the

replacement of a CH fragment by nitrogen decreases aromatic-

ity. According to the classification pattern of the SOM, benzene

and pyridine are the most aromatic in this series, followed by

diazines and, finally, triazine. It has been stated that, for rings

with the same number of nitrogen atoms, the most aromatic are

those having the largest number of N��N bonds.65 However, we

observe pyrazine (146) � pyridazine (148) [ pyrimidine (147)

failing only in the ordering of pyrazine (Table 2D). It is impor-

tant to note that the expected order is not followed by any of

the 10 indices analyzed by Sola and coworkers60 and, therefore,

the initial presumption should be revisited. Interestingly, the ring

size dependence of the NICSzz(1) indices is avoided using our

aromaticity scale dj, which shows that benzene and pyridine are

more aromatic than any of the five-membered rings considered.

The following set analyzes the aromaticity of several mono-

substituted benzenes (Table 2E). Krygowski et al. have showed

that the substituents influence only very weakly the p-electrondelocalization in the ring.66 The major changes in the geometry

of 74 monosubstituted benzenes concern the ring angles and

they are related to the electronegativity of substituents.67 Only

substituents CH21 and CH2

2 cause substantial bond length

changes and HOMA decreases to � 0.7.68 Moreover, all sub-

stituents induce a loss of aromaticity independently of their elec-

tron donating or accepting character. SOM places all the ben-

zene derivatives in the neuron which represent the highest aro-

maticity indicating high resistance of the p-electron to the

substituents effect. A larger map is necessary to distinguish com-

pounds with very similar degrees of aromaticity. An extension

of the region of the highly aromatic compounds with dj between0 and 10, obtained from a 14 3 12 map training only with aro-

matic compounds, is shown in the Figure 3b. This larger map is

able to discriminate benzene, pyridine, and the substituted ben-

zenes, indicating correctly that benzene is the most aromatic

(144, dj 5 0.0). All the benzene derivatives possess dj valuesranging from 1.9 to 5.1, being the N2

1 and NH2 substituents that

lead to the largest decrease in aromaticity in agreement with

other indicators of aromaticity such as para-delocalization index,

aromatic fluctuation index, and multicenter indices.60

Finally, we have analyzed the dependence of the Euclidean

distance on the size of the ring (Table 2F). For this purpose, we

evaluated the aromaticity of three 6p-electron systems with dif-

ferent ring sizes: benzene, cycloheptatrienyl cation (C7H71), and

cyclooctetraenyl dication (C8H821). As the ring size increases it

is expected that the aromaticity decreases, giving the following

order of aromaticity: benzene [ C7H71 [ C8H8

21. This is

indeed the order provided by the Euclidean distance. To distin-

guish between benzene and C7H71, we employed the larger map

shown in Figure 3b to estimate dj. It is remarkable that the posi-

tion of benzene in the bidimensional map does not change using

different values for the aromatic stabilization energy and the ex-

altation calculated from homodesmotic/isodesmic reactions (Sup-

porting Information Table S3). Therefore, the neural network is

hardly sensitive to the reactions employed for evaluating the

energetic and magnetic descriptors of aromaticity. It is noticed

that two methods have been recently reported in the literature

which evaluates stabilization energies without using homodes-

motic/isodesmic equations. We are regarding the possibility of

including these descriptors in future developments of the

SOMs.23,69,70

Additionally, NICS scans have been calculated for borderline

compounds of the three classes (aromatic, nonaromatic, and anti-

aromatic) in order to study in depth the presence of diamagnetic

and paramagnetic ring currents (Fig. 4). The NICS scan proce-

dure is based on scanning NICS values over the distance and

separating them into in-plane and out-of-plane contributions.52

The shape of the NICS scans for benzene (144) and 2,3,4,5-tet-

rahydroxycyclopentadienyl anion (117) is typical for aromatic

systems, showing a minima for the out-of-plane component and

isotropic values which points out the presence of diamagnetic

ring current. The minimum value for the out-of-plane compo-

nent is observed at r 5 1.0 A, although the magnitude of

NICSzz decreases around 9 ppm from benzene to 117, indicat-

ing a smaller diamagnetic ring current. It is noteworthy that the

diamagnetic contribution is larger than the paramagnetic

contribution to the out-of-plane component in both compounds.

However, in 3-phospaphosphole (12) and gallole (47), classi-

fied as nonaromatic compounds by the neural network, the dia-

magnetic contribution is larger than the paramagnetic contribu-

tion to the out-of-plane component at short distances, becoming

equal at 1.3 and 2.7 A, respectively. As Euclidean distance

increases, the minima of the out-of-plane curve become less

negative and appear at larger distances from the molecular

plane, indicating an enhancement of the paramagnetic contribu-

tion and a smaller diatropic ring current. This minimum disap-

pears in the NICS scan curves of the antiaromatic compounds

129 and 35 because the paramagnetic contribution dominates

over the diamagnetic counterpart in the out-of-plane component.

Additionally, the NICS scans shows that the isotropic curves are

governed by the out-of-plane component in the aromatic and

antiaromatic compounds and about equally by the in-plane and

out-of-plane components in the nonaromatic systems. Therefore,

on the basis of the NICS scan results for borderline compounds,

we conclude that the neural network classifies correctly an

extensive dataset of organic compounds into aromatic, nonaro-

matic, and antiaromatic.

Conclusions

The results presented in this article demonstrate that the neural

network fulfils the expected aromaticity trends based on the

chemical knowledge. In addition, the Euclidean distance

between neurons (dj) indicates the expected order of aromaticity

of penta- and heptafulvenes, heteroaromatic species, substituted



benzenes, and substituted cyclopentadienyl compounds. More

important is that the Euclidean distance values (dj), as a measure

of aromaticity, overcomes the limitations of the structural, mag-

netic, and energetic indicators of aromaticity such as the ring

size dependence of L and NICS or the dependence of the ASE

values on the formulation of the reaction scheme.

As a conclusion, we can state that the Euclidean distance (dj)between neurons in the SOM is the first aromaticity scale reported

in the literature which contains the energetic, structural, and mag-

netic aspects of this phenomenon. The results reported for a wide

dataset of p-organic compounds clearly show that the multifac-

eted nature of aromaticity can be only apprehended by a multidi-

mensional method. Additionally, the quantification of aromaticity/

antiaromaticity through neural networks has several advantages.

In contrast to linear regression models, the trained SOM can be

used to classify new compounds and predict its degree of p-elec-tronic delocalization with a short computational time. Further-

more, the position of a new compound on the bidimensional map

is conveniently visualized, indicating successfully the following:

a. the group (aromatic, nonaromatic, or antiaromatic) to which

the new system belongs;

b. their degree of aromaticity/antiaromaticity based on the Eu-

clidean distance;

c. the similarity in aromaticity between different compounds.

On the other hand, a two-dimensional map can much better

reflect the results of the various influences on the aromaticity/antiaro-

maticity: different directions in the map represent different kinds of

similarity and different distances indicate distinct levels of similarity.

Work is underway to expand the present methodology to

other kinds of chemicals as well as to unveil structure–property

relationships.

Acknowledgments

This work was taken in part from the Ph.D. thesis of M.A. The

authors thank Dr. A. Chana for helpful discussions on neural

networks, and Dr. J.D. Guillen for helpful assistance on statisti-

cal analyses. The authors also thank the Centro de Supercompu-

tacion de Galicia (CESGA) for computational time at the SVGD

supercomputer.

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A universal scale of aromaticity for π-organic compounds