Quantifying aromaticity with electron delocalisation measures

Aromaticity cannot be measured directly by any physical or chemical experiment because it is not a well-defined magnitude. Its quantification is done indirectly from the measure of different properties that are usually found in aromatic compounds such as bond length equalisation, energetic stabilisation, and particular magnetic behaviour associated with induced ring currents. These properties have been used to set up the myriad of structural-, energeticand magnetic-based indices of aromaticity known to date. Cyclic delocalisation of mobile electrons in two or three dimensions is probably one of the key aspects that characterise aromatic compounds. However, it has not been until the last decade that electron delocalisation measures have been widely employed to quantify aromaticity. Some of these new indicators of aromaticity such as the PDI, FLU, ING, and INB were defined in our group. In this paper, we review the different existent descriptors of aromaticity that are based on electron delocalisation properties, we compare their performance with indices based on other properties, and we summarise a number of applications of electronic-based indices for the analysis of aromaticity in interesting chemical problems. Page 1 of 54 Chemical Society Reviews


Introduction
The study of aromatic species goes back to 1825 when Michael Faraday obtained benzene by distillation and named it "dicarburet of hydrogen". In this work, Faraday found that the empirical formula of benzene was CH. He already noted that "dicarburet of hydrogen" was much less reactive than "monocarburet of hydrogen" (trans-2-butene). 1 Such decreased reactivity was considered an experimental characteristic of aromatic compounds ever since. In 1834, Mitscherlich determined that the compound synthesised by Faraday had the molecular formula C6H6. 2 Since it was obtained from distillation of benzoic acid (from gum benzoin) and lime, he named the compound as benzin, which became benzene when translated into English. In 1865, a century and a half ago, 3 August Kekulé 4 proposed the molecular structure of benzene consisting in a sixmembered ring (6-MR) of carbon atoms with alternating single and double bonds. Pyridine, the first heteroaromatic compound, was synthesised by Thomas Anderson 5 in 1868 through studies on the distillation of bone-oil and other animal matter. In 1911, Willstätter and Waser 6 synthesised cyclooctatetraene, an eight-membered carbon ring with alternating single and double bonds that had a reactivity very different from benzene and was classified as antiaromatic. The first synthesised inorganic heteroaromatic compound was borazine, B3H6N3, 7 obtained in 1926 by a reaction of diborane with ammonia. In 1954, Doering and Knox 8 prepared the tropylium cation, C7H7 + , which was considered the first verification of the Hückel rule. 9 Four years later, Winstein introduced the homoaromaticity concept while studying the 3-bicyclo[3.1.0]hexyl cation. 10 Three years later, the synthesis of the first organic derivatives of closododecaborate and closo-decaborate by the group of Muetterties 11 was the beginning of closo-borane chemistry and three-dimensional aromaticity, the type of aromaticity that characterises fullerenes. 12 The identification 13,14 of the planar triplet ground states of C5H5 + and C5Cl5 + in the late 60's provided experimental support for the existence of triplet aromaticity as predicted by Baird. 15 In 1982 Roper et al. 16 synthesised the first metallabenzene, an osmabenzene, thus initiating a new group of aromatic species, the so-called metalloaromatic compounds. 17 Interestingly, metallabenzene species were proposed first by Thorn and Hoffmann three years before from theoretical calculations. 18 More recently, in 2001 Boldyrev, Wang, and coworkers 19 detected a series of bimetallic clusters containing Al4 2-, the first all-metal aromatic cluster known, face-capped by an M + cation (M = Li, Na, Cu). The same group detected Ta3O3 -, the first discovered metallic cluster showing δ-aromaticity. 20 In 2003, Herges et al. synthesised the first Möbius aromatic hydrocarbon. 21 This kind of aromatic species were already predicted by Heilbronner forty years before on purely theoretical grounds. 22 Despite the nearly two centuries of intense study and continuous progress described in the previous paragraph, the interest on aromaticity has not decreased and still stimulates the creativity of a number of contemporary chemists. Considered like a chemical unicorn by Frenking and Krapp,23 aromaticity is not a property directly observable and it lacks a well-founded physical basis. Therefore, its definition and quantification remains elusive. Chen and Schleyer 24 defined aromaticity as "a manifestation of electron delocalisation in closed circuits, either in two or three dimensions." Electron delocalisation is without doubt one of the key aspects of aromatic compounds. Since 1932 when Pauling 25,26 introduced the concept of resonance, it is well established that delocalisation of electrons in a molecule can stabilise it. Another relevant feature of aromatic compounds is their symmetry. Despite not all aromatic compounds are symmetric and not all symmetric cyclic compounds are aromatic, it is generally the case that the most archetypal aromatic compounds are highly symmetric and possess degenerate highest-occupied molecular orbitals that are fully occupied resulting in a closed-shell structure or have a same-spin half-filled electronic structure. This is the case of benzene, but also of triplet C5H5 + , C60 10+ , Al4 2or closo borane clusters like B6H6 2-. The closed-shell or same-spin half-filled electronic structure is the origin of several rules of aromaticity such as the 4n+2 Hückel, 9 4n Baird, 15 2n+2 Wade-Mingos, 27, 28 2(n+1) 2 Hirsch 29 or the 2n 2 +2n+1 30 rules. This particular electronic structure of aromatic species explains their substantial energetic stabilisation and, consequently, their low reactivity.
Moreover, it results in a variety of unusual chemical and physical properties, including tendency toward bond equalisation, unusual reactivity, and characteristic spectroscopic and magnetic features.
It is worth noting that the bond length equalisation observed in benzene and derivatives is enforced by the σ-electrons and not by the π-electron system. The latter is distortive and favours the D3h structure of benzene over the D6h one.
This somewhat paradoxical observation was suggested first by Longuet-Higgins and Salem in 1959. 31 Berry 32 used this suggestion to account for the observed increased frequency of the b2u Kekulé vibrational mode when going from the ground 1 A1g to the first 1 B2u excited state. The same idea was reinforced later on by the work of Haas and Zilberg, 33,34 and especially by that of Hiberty, Shaik, and co-workers 35,36 and others 37,38 Finally, more recently Pierrefixe and Bickelhaupt [39][40][41] showed that the regular geometry of benzene is a consequence of how σ and π overlaps depend on bond distances. In aromatic annulenes the authors confirmed that bond equalisation is due to the σ electrons whereas the π-electron system favours doublebond localisation.
The quantification of physicochemical properties that reflect some manifestation of the aromatic character of molecules is used to evaluate their global or local (e.g., individual rings in a polycyclic arene) aromatic character. [42][43][44] This leads to the countless existing measures of aromaticity based on the structural, 45,46 magnetic, 24,47 energetic, 48 and electronic 49 properties of molecules. These indicators provide indirect measures of aromaticity that, to some extent, are somewhat arbitrary. Moreover, for a series of compounds different descriptors of aromaticity do not provide the same aromaticity ordering and some descriptors fail to correctly quantify certain changes of aromaticity in particular examples. 50 For this reason, it is widely accepted that the concept of aromaticity should be analysed by employing a set of aromaticity descriptors. 24,51,52 The importance of electron delocalisation in aromatic species is universally recognised. It is thus reasonable to employ electron delocalisation as a tool to construct new aromaticity indicators. The problem is that electron delocalisation like aromaticity is not an observable and, therefore, there is not a unique way to measure it. Moreover, when devising new aromaticity descriptors three steps should be followed: development, assessment, and application. In the next sections, we describe firstly the most usual ways to quantify electron delocalisation; second, we discuss how these measures can be used to define indicators of aromaticity; third, we examine how electronic-based indices compare with other existing descriptors; and finally, we show some applications of these electronic descriptors of aromaticity carried out in our research group.

How can electronic delocalisation be measured?
Electron localisation/delocalisation is central in several fundamental chemical phenomena such as conjugation, hyperconjugation, and aromaticity that are important to explain the structure, stability, magnetic properties, and reactivity of many molecules. Several tools have been developed to quantify delocalisation in a molecule and to provide new insights into chemical bonding. These descriptors of delocalisation have been reviewed in several works. 49,[53][54][55] We just briefly mention here some of the most important and describe in more detail those utilised for the definition of aromaticity indicators.
Descriptors of electron localisation/delocalisation can be broadly grouped into three classes, namely, those computed from the wavefunction, those constructed directly from the electron density and its derivatives, and those that are derived from first-and higher-order density matrices. Among the first group of electron delocalisation indicators, we can mention the use of weights of resonance structures and energies in valence bond theory. 56,57 Also belonging to this group are the measures of delocalisation obtained with the block-localised wavefunction (BLW) method of Mo and coworkers. 58,59 In addition, in the framework of the molecular orbital (MO) theory, different techniques are available for the localisation of MOs to find the regions where electron pairs are located. To date the natural bond orbital (NBO) analysis of Weinhold et al. 60,61 is one of the most used method for localising bonds and lone pairs. The importance of electron delocalisation is assessed in NBO analyses by approximated second-order perturbative expressions. Another similar tool used for obtaining patterns of chemical bonding is the adaptive natural density partitioning (AdNDP) method developed by Zubarev and Boldyrev. 62,63 AdNDP represents the electronic structure in terms of n centre -2 electrons (nc-2e) bonds. Starting from n = 1, AdNDP recovers Lewis bonding elements (1c-2e and 2c-2e objects) and delocalised bonding elements (for n > 2), which can be associated with the concepts of delocalisation and aromaticity. An even more recent method is the orbital localisation procedure based on the electron localisation function (ELF-LOC) of Alcoba, Tiznado, and coworkers. [64][65][66] This procedure localises the molecular orbitals in regions that have the highest probability for finding a pair of electrons, providing also a chemical bonding description in terms of nc-2e bonds.
In the second group of electron delocalisation descriptors, we can refer to the Laplacian of the electron density (∇ 2 ρ(r)), 67, 68 the ellipticity along the bond path (ε=λ1/λ2-1), 69 the noncovalent interactions index (NCI), 70 the inhomogeneity measures of the electron density, 71 the source function, 72, 73 the single exponential decay detector, 74 and a recent electron delocalisation index obtained from electron population analysis. 75 The electron delocalisation indicators collected in the third group are probably the most abundant. The reason is that the location of electron pairs involves two spatial coordinates and, therefore, methods based on functions of two (or more) positions such as the first-order density matrix and the two-electron density or pair density are more suitable to analyse electron localisation and delocalisation. This group includes the localised orbital locator, 76 the electron delocalisation range function, 77 the   parity function, 78 the analytical method by Proynov to calculate the population of  effectively unpaired electrons, 79 the Fermi hole density maps, 80, 81 the domainaveraged Fermi holes, 82, 83 the Laplacian of the exchange-correlation density, 84 the electron localisation function (ELF), [85][86][87][88] and the electron localisability indicator. 89,90 Also belonging to this group are the methods for computing the probability of finding a certain number of electrons in a given volume. [91][92][93] And finally, although not derived directly from density matrices, the analysis of the linear response kernel (LRK), 94-96 a function of two position variables, could also be included in this third group of electron localisation/delocalisation descriptors.
Because of its importance in the definition of aromaticity descriptors, we will refer with more detail to the electron sharing indices (ESI). These ESIs are electron localisation/delocalisation descriptors that can also be classified in the third group. They are defined from the spinless two-electron density or pair density, ߛሺ‫ݎ‬ ଵ ሬሬሬԦ, ‫ݎ‬ ଶ ሬሬሬԦሻ. This function can be interpreted as the probability density of therefore, the following sum rule is followed: Eqs.  The definition of the ESI can be generalised to analyse multicentre delocalisation or sharing of electrons. It is possible to define a multicentre DI or Mc-ESI 106 between the M centres A1 to AM: where ni are the occupancies of the natural orbitals. This formulation assumes that the electron sharing occurs only between neighbouring atoms. Bultinck and coworkers 107  One should also mention another important pitfall of multicentre indices: the atomic partition. Both IA1,···,AM and MCI are very sensible to the atomic partition, being QTAIM and TFVC partitions the most reliable ones. 109 In addition, the accuracy of the numerical integration over the atomic basins becomes an issue for large strings of atoms, as those occurring in molecules with large rings.
In this sense, rings of more than ten members need very accurate calculations or the use of the Hilbert-space partition that provides analytical atomic overlaps, thus avoiding the integration hassle.

Electron delocalisation measures as indicators of aromaticity
In this section we will describe the electron delocalisation aromaticity indices and give the expression of the most important ones. In order to illustrate the performance of these indicators we will include the values for a small set of representative molecules: benzene, cyclohexane, borazine, pyridine, Al4 2-, and the transition state of the Diels-Alder cycloaddition reaction between butadiene and ethylene that we have calculated at the CCSD/cc-pVDZ level of theory. We  115 Electron delocalisation has been tightly connected to aromaticity from the very beginning. The old concept of bond order -that later evolved to electron sharing index, Eq. (3)-was brought up by Coulson in 1939 and it was first applied within the Hückel molecular orbital method to study the electronic structure of aromatic molecules. 116 Julg was among the first to suggest an index based on the uniformity of the interatomic distances, an indicator that a few years later he modified to take into account the charge gradient between bonded atom pairs in a ring. 117,118 In 1983, Jug suggested that aromaticity could be measured by the minimal bond order in a given ring structure 119  with respect to some aromatic reference, and it is given by: 122 where A0 ≡ AM , the atomic delocalisation is defined as: and α is a simple function to make sure that the first term in Eq. (9) is always greater or equal to 1, The CC and CN bonds reference values are taken from benzene and pyridine in its ground state. FLU is close to 0 in aromatic species and increases as the molecule departs from the aromatic reference. Like any aromaticity index based on reference values it depends critically on the model aromatic molecules chosen and it cannot be used to study reactivity. 123 Moreover, it is difficult to compare molecules with different ring patterns. However, FLU gives a good account of the aromaticity for ground-state organic molecules. 124 These facts are illustrated by the numbers in Table 1 Table 1 The CCSD/cc-pVDZ electronic-based aromaticity indices for a set of representative molecules and the transition state of the Diels-Alder cycloaddition (TS-DA). With the exception of FLU, for all indicators the larger the index, the more aromatic the species are. Values multiplied by 1000.
The para-delocalisation index (PDI) measures the electron delocalisation across the ring by averaging the three para-related positions in a 6-MR. 125 PDI is, obviously, limited to rings of six members and it suffers to describe molecular rings containing atoms with lone-pairs (it finds pyridine more aromatic than benzene) or to explain the small distortions around the equilibrium geometry (see section 4). 50,126 On the other hand, it does a good job in polycyclic aromatic hydrocarbons. 127 Ángyán 128, 129 showed the connection between the 2c-ESI and the LRK for the QTAIM partition, which suggests that the PLR and the PDI could be measuring the same effect in a molecular ring.
The Iring was defined by Giambiagi and coworkers 130 as their own multicentre index, Eq. (7), applied to ring structures, assuming the obvious link between cyclic electron delocalisation and aromaticity. The MCI of Bultinck can also be used to quantify aromaticity. Unlike Iring, which only takes into account the Kekulé arrangement of the atoms in a ring, MCI considers all possible arrangements. The Kekulé structure contribution is usually the most important one and, therefore, Iring and MCI rarely give disparate results. In such eventuality, the ring cross-contributions play a prominent role, as it happens in Al4 2-, where the delocalisation between non-bonded atoms is half as large as the delocalisation of neighbouring pairs. Unlike PDI and FLU, these multicentre indices do not have severe restrictions that limit its range of application, except for the numerical accuracy problems already mentioned. However, both Iring and MCI multiply a number of overlaps that depend on the ring size, leading to a sizeextensivity problem. Since overlap values are lower than one in absolute value, the MCI becomes artificially large for small rings, as we can see in the case of Al4 2-, which has an MCI value an order of magnitude larger than benzene. The latter issue was recently solved in our group by taking the Mth root of these two indices. 131 In addition, we also proved that taking into account the appropriate normalization factor these indices showed a good linear correlation with the topological resonance energy per π electron (TREPE). The normalised Iring and MCI values were named ING and INB, 131 their expression reading: where N π is the number of π electrons of the atoms in the ring, G(N π ) equals 1 and -3 for aromatic and antiaromatic systems, respectively, and A longstanding goal within the field of aromaticity has been the classification of molecules into different groups (aromatic, non-aromatic, antiaromatic) to rationalise structure, reactivity, and molecular properties. This classification was usually done on the grounds of chemical intuition or by examination of the occupied MOs. The discovery of new and more exotic molecules makes this classification more and more complicated, and for the past twenty years several aromaticity indices have been constructed to simplify this task. However, there has been such a proliferation of indices (often giving opposite answers) that is barely impossible to unambiguously characterise new molecules. To gain insight into the nature of aromatic molecules or to design and discover new molecules, aromaticity indices need to be validated.

Assessing the performance of aromaticity indices using a test set
A central tenet in the study of aromaticity is that it cannot be unambiguously The aim of this section is to propose a simple protocol for assessing the performance of existing and prospective aromaticity descriptors and to provide a list of recommendations that may serve as a guide for future studies involving more complex systems.
In a series of papers we analysed whether a descriptor is able to properly account for the trends of aromaticity in a variety of well-established cases. For example, it is clear that benzene is more aromatic than toluene and a good descriptor of aromaticity should capture this fact. Since the development of the first descriptors, authors focused on identifying failures of descriptors in particular cases. However, there is a need for a more robust methodology to assess the performance of an index. The use of test or training sets comprising a number of representative molecules is of common practice in other theoretical chemistry fields, such as density functional theory development, to judge the validity and range of applicability of a new method. In 2008, we proposed to extend this idea to the field of aromaticity suggesting the use of a test set as a tool to assess the performance of aromaticity descriptors. 50,139 To this end, we proposed a series of tests with well-known trends that can be used to appraise the performance of existing and new indices of aromaticity (see Fig. 1). In this sense we are not strictly comparing indices between them but we are analysing their behaviour in a variety of cases that account for situations found along the spectra of aromatic molecules.  Atom size dependence. Since the very beginning, aromaticity rapidly broadened its realm to atoms other than carbon. Atoms of different type are commonly found to form rings that display typically aromatic properties.
Substituting carbon for other atoms perturbs the π-delocalisation patterns of such molecules with respect to benzene. Therefore, by substituting carbonhydrogen fragments by nitrogen atoms to form a chair-like N6 ring one expects a reduction of the aromatic character. All multicentre indices and FLU account for the decrease of aromaticity when going from C6H6 to N6. On the contrary, PDI value increases significantly from 0.103e in benzene to 0.131e in N6, showing PDI is quite dependent on the atom nature. The same trend is observed for NICS(1) and NICS(1)zz, which assign a higher aromatic character to N6, revealing the atom-size dependence of these descriptors. Structure-based HOMA index gives the proper results. However, FLU and HOMA are limited by available reference values, which significantly restrict their applicability to rings containing elements other than carbon. In general, we recommend the use of multicentre indices to study rings with a variety of elements including metalloaromatic systems such as metallacycles and all-metal clusters. These indices shall aid the design and characterization of all-metal clusters with specific properties (see section 5.2).

Heteroaromatic series, Clar's systems, and fulvene series.
To complete the proposed test set with other relevant situations displayed by aromatic molecules, we included a total of three more tests. First, we introduced a test to predict the proper trend of aromaticity along the following well-established heteroaromatic series C4H4X (X= CH -, NH, O, CH2, BH, CH + ). Second, we proposed a test that analyses the effect of fusing aromatic rings represented by five Clar systems. Third, we included a test to assess the expected trend of aromaticity in 5-MR and 7-MR fulvenes with different substituents. In general, all electronic-based indices passed the three tests (see Fig. 1). The only exception to properly account for the predicted order of aromaticity in a series of fulvenes is all-metal 152,153 and semimetal clusters, etc. has prompted a revolution in the study of aromaticity. 17 At variance with the classical aromatic organic molecules that only possess π-electron delocalisation, these compounds have σ-, π-, δ-and φ-electron delocalisation, which can be even combined to give double or triple aromaticity, the so-called multifold aromaticity. This latter issue makes most of the classic indicators of aromaticity not valid to discuss these complex systems, 49,50 and for such reason more general and reliable indices of aromaticity are needed. These indices must fulfil two requirements: first, to be free from reference values (since it is difficult to choose a most aromatic reference molecule); and second, indices should be separable into σ-, π-, δ-, and φcomponents of aromaticity. Multicentre electronic delocalisation indices and NICS are among the few indicators that fulfil these two requirements for rings of arbitrary size. 139,154 The new aromaticity test consists of the 4-MR series of valence isoelectronic species [XnY4-n] q± (X, Y = Al, Ga, Si, and Ge; n = 0-4), which should exhibit particular aromaticity trends. 154 Let us focus on the series from Al4 2to Ge4 2+ ; similar trends are expected for the other series. We anticipate a steep decrease in aromaticity when going from Al4 2to Al3Gedue to the reduction of symmetry and to the substitution of one Al atom by a more electronegative Ge atom. Although more arguable, we anticipate a smooth aromaticity reduction going from Al3Geto Al2Ge2. The same decrease should occur from Ge4 2+ to Al2Ge2; giving the following trend of aromaticity: Al4 2-> Al3Ge -≥ Al2Ge2 ≤ AlGe3 + < Ge4 2+ . From Fig. 2, it is observed how both total MCI and its π-component (MCI π ) successfully provide the expected order of aromaticity, showing a concave U shape. NICS indices were submitted to the same test but only NICS(0) π RCP provided the expected trend. In these rings, with different atomic sizes, the calculation of NICS should be performed at the ring critical point (RCP), as Page 28 of 54 Chemical Society Reviews suggested by Morao and coworkers. 155 The different analogous series analysed yield similar conclusions. Therefore, we concluded that electronic indices are also suitable to analyse aromaticity in all-metal and semimetal clusters.  Figs. 1 and 2. We hope that the results of this test set will provide a set of guidelines that will drive the choice of aromaticity indicators in future studies and will lead to the construction of more robust and applicable descriptors.

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Chemical Society Reviews

Some applications
In this section, we present some of the most relevant applications carried out by our research group using the above defined electronic-based measures of aromaticity.

Polycyclic aromatic hydrocarbons and derivatives
In our first application, a set of planar and bowl-shaped PAHs, together with C60 and C70 fullerenes, were evaluated by means of PDI and FLU indices, showing that these indices could identify regions of local aromaticity and antiaromaticity in PAHs and fullerenes. PDI and FLU values for C60 and C70 indicated the relatively weak local aromaticity of the 6-MRs and the nonaromatic or antiaromatic character of the 5-MRs in fullerenes. 125,127,156 Acenes, phenacenes, and non-planar helicenes are three different series of benzenoid compounds. Using FLU and PDI 157

Substituent effects on aromaticity.
The analysis of the substituent effects on benzene proved the high resistance of aromatic systems to disrupt the π-electron structure in electrophilic aromatic substitution reactions. In particular, changes of PDI when going from benzene to substituted benzene derivatives are small and correlated with Hammett substituent constants. 142 The same behaviour was observed for the complexation of a lithium cation to a series of PAHs. 159 Substituent effects were also studied in 4-substituted-1,2-benzoquinones. Results show that only the keto group in meta position is affected by the electron-donating/attracting power of the substituent, whereas the para-related C=O is not. Although MCI and FLU display small changes, these indices agree on assigning a more aromatic character to rings with electrondonating substituents. 160 On the other hand, the substituents do have a large effect on the aromaticity of pyrazoles and imidazoles with N-substituents. 161 We found that the imidazole ring is more stable than the pyrazole one. The reason for the relative energy difference was attributed to the weakness of the NN bond in the latter, and not to a higher aromaticity of the former, as both rings present similar MCI and FLU values. 162 By comparison to the corresponding substituted series of benzene, this latter appears not only to be more aromatic, but also more robust towards substitution effects as denoted by the slope of the FLU 1/2 vs. σR correlation (see Fig. 3). A recent work showed that 1-indenones and their aza derivatives are more stable than 2-indenones because their 6-MR is more aromatic. 163 Interestingly, tetrafluorination of the 6-MR in such compounds hardly causes any change in the local aromaticity of this ring, thus confirming that the aromaticity of benzene rings is quite robust.

Discrepancy with magnetic measure of aromaticity.
Pyracylene was the first case in which we showed that the calculation of NICS at the ring centre should be analysed with caution, as NICS(0) wrongly assigned an increase of the local aromaticity of 6-MRs upon distortion from planar to pyramidalised pyracylene. 164 In contrast, NICS(1) calculated 1 Å above pyracylene reported the expected decrease of aromaticity upon bending. Both PDI and ring currents contradicted NICS, giving the expected reduction of aromaticity with bending.
Another conflicting case of NICS' performance is [2,2]paracyclophane, in which the NICS predicted decrease of local aromaticity of the stacked rings was not real, but caused by the coupling of the magnetic fields generated by these two rings. 165 The same problem was experienced when the local aromaticity of a series of polyfluorene compounds with increasing number of π-stacked layers was analysed. 166 NICS shows a spurious increase of aromaticity that is due to the coupling between the magnetic fields generated by the π-stacked rings, whereas PDI, FLU, and HOMA showed that aromaticity does not change due to π-stacking. Such interaction causes a change in the strength of the hydrogen bonds (N1-H···N3 and N2-H···O2 become stronger, whereas O6···H-N4 weakens). 168 In turn, these alterations affect the aromaticity of the 5-and 6-MRs of the nucleobases.
The observed increase of the aromaticity of the guanine and cytosine 6-MRs due to the interaction with Cu + and Ca 2+ was attributed to the strengthening of hydrogen bonding in the guanine-cytosine pair that stabilises the resonance structure with a π-sextet in the 6-MRs. On the other hand, the reduction of aromaticity in the 5-and 6-MRs rings of guanine due to interaction with Cu 2+ is caused by the oxidation process, which removes a π electron disrupting the π- systems to aromatic (4n+2)π systems. The change in π-electronic delocalisation when we move from a (4n+2)π-aromatic to a 4(n+1)π-antiaromatic species by adding a pair of electrons is much smaller (see Fig. 4). FLU, PDI, and MCI criteria, as well as the cross terms of the total π-electron delocalisation, correctly assign an aromatic or antiaromatic character to each system, regardless the number of electrons. Therefore, these indices provide aromaticity values in agreement with Hückel's rule. 173,174  aromatic. 30 For instance, neutral C60 appears to be non-aromatic according to MCI and NICS(1)zz; whereas singlet C60 10+ that obeys the 2(n+1) 2 rule, is found to be aromatic. On the other hand, C60 19+ with S=9/2 and C60 1with S=11/2 following the 2n 2 +2n+1 rule, appear to be even more aromatic than C60 10+ . This new rule may become a powerful tool to study the stability of high-spin spherical molecules. In section 5.3 we comment on the aromaticity of excited states in connection with the Baird rule.

Aromaticity determines the regioselectivity of Diels-Alder
reactions in fullerenes. The reactivity and regioselectivity of Diels-Alder (DA) reaction involving empty fullerenes is generally favoured for [6,6] bonds, whereas in endohedral metallofullerenes [5,6] bonds are commonly more reactive. 175,176 When a metal cluster is encapsulated inside a fullerene there is a charge transfer from the metal cluster to the fullerene. In the case of M3N units (M = Sc, Y, Gd…) formally six electrons are transferred to the carbon cage. We decided to analyse the effect of adding electrons to the fullerene cage by calculating the reaction profile for the DA reaction of cyclopentadiene (Cp) to the [6,6] and [5,6] bonds of C60 n-(n = 0 -6) species. 176 The C60 n-(n > 0) was taken as a model for the cage of endohedral metallofullerenes. It was found that the reaction becomes more exothermic (and the barrier is reduced) for the [5,6] attack when n increases from 0 to 6 electrons (see Fig. 5). On the other hand, for the [6,6] addition the exothermicity is somewhat reduced (and the barrier increases) when n increases. For n = 4-5, there is a change in the regioselectivity of the process and the [5,6] becomes the preferred attack. To understand the change of regioselectivity in C60 upon reduction, we calculated the MCI for the 5-and 6-MRs of C60 nshowing that the aromaticity of the 5-and 6-MRs increase and decrease, respectively, with the successive addition of electrons to the C60 molecule (see Fig. 5). Because the DA addition leads to a change from planar sp 2 to tetrahedral sp 3 for the C atoms of the attacked bond, when the addition occurs on a [6,6]-bond type, the π-conjugation of two 5-MRs and two 6-MRs is lost. On the other hand, when the addition is on a corannulenic [5,6]-bond, the conjugation vanishes in three 6-MRs and one 5-MR. For neutral C60, the preferred attack is the [6,6] addition because the aromaticity of only two of the most aromatic 6-MRs is lost (the [5,6] attack affects the aromaticity of three 6-MRs). On the other hand, for C60 6the most favourable addition is at the [5,6] bond because the aromaticity of only one of the most aromatic 5-MRs is lost.
In this case, the changes in aromaticity between a 6-MR and a 5-MR upon reduction determine the regioselectivity of the DA additions to fullerenes. 176

Metalloaromaticity
One of the most important findings in metalloaromaticity was the discovery of the Al4 2cluster by Boldyrev, Wang, and coworkers. 19 This cluster was the first example of all-metal species with σ-and π-aromaticity. It is well known that π-electrons in benzene are distortive and it is the σ-skeleton the responsible for the D6h symmetry of benzene. We wondered whether the σ-and π-electrons in Al4 2favoured the D4h or the D2h structure. To solve this question, we performed an energy decomposition analysis (EDA, see Fig. 7) showing that the π-electrons in Al4 2prefer the D2h structure but the σ-electrons force the double bond to delocalise, leading to the regular D4h geometry. 153 This analogue behaviour to benzene was explained through the corresponding MO diagram (see Fig. 8), where it is observed how both the π and the radial-σ (σR) orbitals induce distortion, whereas the tangential occupied orbitals (σT) are the ones responsible for the D4h structure. This different character of σR and σT orbitals is also supported by MCI indices that show a more important contribution to the total σ-aromaticity from the radial than from the tangential orbitals.   NaMg3and Na2Mg3 clusters containing the cyclo-[Mg3] 2unit are two of the very few electronic species with π-bonding without the occurrence of a σframework. 181,182 Interestingly, the aromaticity switches from σ to π when the Mg3 2unit coordinates Na atoms to give NaMg3and Na2Mg3 clusters. The distance between the coordinated Na atom and Mg3 can be used to tune the aromaticity and prompt an unprecedented switch from σ-to π-aromaticity.
The aromaticity of all-metal clusters with transition metal complexes involving d and f orbitals implies more complicated analyses due to the large number of electrons involved and the inclusion of relativistic effects. The occurrence of highly delocalised valence electrons occupying the large angular momentum orbitals in transition metals, gives rise to multifold aromaticity. To study this phenomenon, we analysed the aromaticity of the series Cu3 + , Y3 -, La3 -, Ta3O3 -, Hf3, 5 Ta3 -, and 3 Hf3 with the MCI index. 183 Cu3 + was confirmed to present exclusive σ-aromatic character; whereas Y3and La3present σ-and π-aromaticity, supported by MCI σ and MCI π values. On the other hand, Ta3O3is the first cluster presenting both π-and δ-aromaticity, as confirmed by the small value of MCI σ , and the twice as large MCI π and MCI δ values. The aromaticity patterns of Hf3 were even more complex, showing prominent σ-, but also significant π-and δ-orbital contributions (the so-called threefold aromaticity).
In a recent work 184

Aromaticity of excited states
In a recent review, Ottosson and co-workers underlined the importance of applying the concept of aromaticity to rationalise excited state properties and reactions. 144 Aromaticity and antiaromaticity effects observed in the excited states may play a similar role in understanding reactivity and molecular properties as in the ground state. To gain insight into the nature of these effects we need to know which ground-state descriptors of aromaticity are transferable to excited states. One of the main difficulties that one encounters is that few descriptors of aromaticity can be easily employed to assess the aromatic character of excited states. Karadakov made the first calculations at the CASSCF level of magnetic aromaticity using NICS and other measures to describe the aromatic character of low-lying singlet and triplet excited states of benzene and related compounds. The results obtained were shown to be in line with Baird's rule. Among electronic-based descriptors only ELFπ has played a major role in studying the aromatic character of lowest-lying triplet states in fulvenes but no attempts have been done to quantify the electronic delocalisation in electronic states of higher energy. 144 To bridge this gap, we proposed to generalise the use of electronic indices, PDI, FLU, Iring, and MCI to study the aromaticity of a set of simple molecules in a number of excited states. 185 To this end, by means of DFT and CASSCF calculations, we studied the aromaticity patterns of the low-lying singlet, triplet, quintet, and septet excited states of benzene, cyclobutadiene, and Fowler to account for high-order multiplicities such as quintet and septet states. 186 That is, compounds with (4n+2)π-electrons that are aromatic in their lowest-lying singlet state should also be aromatic in their lowest-lying quintet state and antiaromatic in their lowest-lying triplet and septet states. On the contrary, molecules with 4nπ-electrons are antiaromatic in their lowest-lying singlet and triplet states and aromatic in their triplet and septet states. These trends are perfectly followed by the Iring and MCI values summarised in Table 2

Conclusions
Over the last decades there has been a remarkable expansion in the number of different types of aromatic systems and in our understanding of aromaticity. The field of aromaticity is in constant evolution and the variety of molecules that present properties related to aromaticity is growing exponentially. It is our opinion that the field of aromaticity has been enriched (and not cheapened as

TABLE OF CONTENTS GRAPHIC
Aromaticity descriptors based on the quantification of electron delocalization are all-round indicators that outperform most of the classical structural-and magnetic-based indices.