Aromaticity and Antiaromaticity: How to Define Them

Abstract

Aromaticity and antiaromaticity are concepts that are often used to explain and predict the physical and chemical properties of cyclic conjugated compounds. They are associated with 4n + 2 and 4n cyclically arranged electrons that are delocalized and mostly localized, respectively. The large number of papers devoted to these concepts, together with two recent conferences on aromaticity (Aromaticity 2018 and 2025, Mexico) that brought together experts from all over the world to discuss aromaticity and antiaromaticity and their applications testify to their importance, but also to the lack of a simple and easily understandable definition. This review highlights the most important manifestations of (anti)aromaticity by considering selected examples from the literature, chosen to provide us with a clearer picture of these two concepts.

1. Introduction

In 1825, by distilling an oily residue, Michael Faraday isolated a colorless liquid with an odor reminiscent of oil gas, which he named “bicarburet of hydrogen” because he believed that the ratio of carbon to hydrogen was 2:1 [1]. Eight years later, the same compound was isolated by German scientist Eilhard Mitscherlich by distillation of benzoic acid, a substance from gum benzoin. He called it “benzin,” but the name was changed to “benzol” at the suggestion of a journal editor [2]. The name “benzene” is used in other languages to avoid the confusion with alcohols whose names end with –ol. In 1855, August Wilhelm Hofmann was the first to use the term “aromatic” to designate a family of acids related to benzene [3]. The empirical formula of benzene (C/H 1:1) [2] indicated an unsaturated molecule, but its chemical behavior was very different from the behavior of other unsaturated systems: while they were reactive, benzene was rather inert. After a day-dream of a snake seizing its own tail, August Kekulé came up with the idea of a cyclic structure of benzene with alternating single and double bonds, which he proposed in 1865 in a paper written in French [4], and a year later in a paper written in German [5]. If so, two ortho derivatives should be possible, but only one was known. To account for this, Kekulé modified his theory and treated benzene as two rapidly interconverting cyclohexatriene structures [6,7,8]. The hexagonal structure of the benzene ring was proven by X-ray analysis of its hexamethyl derivative [9]. However, Kekulé’s description did not explain the reluctance of benzene to react like other unsaturated compounds. In the early 1930s, John C. Slater [10], Linus Pauling [11,12], and George W. Wheland [12] explained the unusual stability of benzene by the resonance between the two Kekulé structures, which leads to energetic stabilization. Around the same time, Erich Hückel used molecular orbital theory to explain the special stability of benzene and provided the basis for the famous 4n + 2 rule of aromaticity [13,14,15,16]. Thus began the story of aromaticity.

2. Many Types of Aromaticity but Only One Effect

Nowadays, we speak about ground [17,18,19,20,21] and excited-state aromaticity [22,23,24,25,26,27,28], spherical [29,30] and three-dimensional aromaticity [31,32], ground [33] and excited-state homoaromaticity [34], Möbius aromaticity [35], ground [36,37] and excited-state all-metal aromaticity [38], σ-aromaticity [39,40,41,42], carbo-aromaticity [43], adaptive aromaticity [44,45,46], double aromaticity (σ and π [47,48], π [49,50,51], and Möbius [52]), and hyperconjugative aromaticity [53,54,55]. Are there so many aromaticities? It seems like a philosophical question. If π electrons are cyclically delocalized, we can say this is π aromaticity, if they are σ electrons, we can call it σ aromaticity, if the molecule is in the ground state, we talk about ground state aromaticity, but if it is in the excited state, we can talk about excited-state aromaticity, and Möbius topology leads to Möbius aromaticity. They all share one effect: stabilization due to cyclic electron delocalization. In a recent paper, Solà and Bickelhaupt used simple particle-in-a-box (PIB) and particle-on-a-ring (POR) models to show that the latter gives a lower first energy level and that higher energy levels come in pairs [56]. The difference between the two mathematical equations for linear (PIB) and cyclic (POR) systems with the same number of particles (electrons) can be considered as an approximation of aromatic stabilization energy (ASE) [56]. So, the essence of aromaticity is lowering of energy. What energy? This is one of the questions that makes aromaticity a “fuzzy” concept [57,58].

3. Aromaticity and Energy

IUPAC defines the aromaticity concept as “cyclic electron delocalization which provides for the enhanced thermodynamic stability (relative to acyclic structural analogues),” and adds that “a quantitative assessment of the degree of aromaticity is given by the value of the resonance energy” [59]. Further, the resonance energy is defined as “the difference in the potential energy between the actual molecular entity and the contributing structure of lowest potential energy” [60]. For benzene, this should be the difference between the Kekulé structure and its real structure.

Regarding aromaticity, the resonance energy (RE) [61] corresponds to energy lowering due to cyclic delocalization of certain types of electrons (π, σ), while thermodynamic stability depends on all types of delocalization (conjugation and hyperconjugation), strain caused by deviation of bond angles, dihedral angles and bond lengths from optimal values, steric repulsion and electrostatic attraction between atoms or groups, the latter including hydrogen bonding. So, in a series of compounds, the most aromatic one is not always the most thermodynamically stable [62,63,64,65,66,67]. In a recent paper on cyclo [2n]carbons, a comparison with [2n]annulenes is given. It is stated that, for example, the planar circular form of [10]annulene is stabilized by cyclic delocalization of 10 π electrons, but does not exist due to the large angular strain [68]. Benzene has no angular strain. It has strain in the σ-electron system due to compression of the σ bonds from the optimal value of 1.52 Å for Csp²−Csp² single bond to 1.40 Å [69,70]. This strain is overcome by the π-electron RE [69,70]. Interestingly, and contrary to what students learn in introductory organic chemistry courses, it is the tendency of the σ-electron system to form a symmetrical structure that allows for the delocalization of π electrons, while the π electrons prefer to be localized [71,72,73].

If we call aromaticity only when cyclic delocalization is an effect that provides for enhanced thermodynamic stability, what do we call it when cyclic delocalization does not provide for enhanced thermodynamic stability? It seems clearer to speak of aromaticity as a stabilizing effect resulting from cyclic electron delocalization, which affects thermodynamic stability, but is not always responsible for its enhancement.

Evaluation of Aromatic Stabilization Energy

To obtain the value of RE, we need to know the energy of the actual structure and the most stable resonance contributor, where the latter is not a viable structure. Shaik and co-workers proposed the Kekulé-crossing model to account for the frequency up-shift of the Kekulé type b_2u mode in the ¹B_2u excited state of benzene and linear acenes. Considering that the electronic ground state and the first excited singlet state come from the in-phase and out-of-phase combinations of Kekulé structures, respectively [74,75], the excitation energy measures the magnitude of the vertical resonance energy (VRE) [69]. However, some delocalization can be expected in the 1,3,5-cyclohexatriene structure. The estimated VRE in 1,3,5-cyclohexatriene with the bond lengths of 1.314 Å and 1.522 Å amounts 40.2 kcal/mol, which is 44% of the VRE of benzene [70].

Experimentally, REs were estimated using the heats of hydrogenation and combustion. However, they suffered from unbalanced conjugation, hyperconjugation, change in hybridization, strain effects, and experimental error [61,76]. Instead of experimentally determined enthalpy of hydrogenation, they could be calculated using experimental enthalpies of formation of (anti)aromatic compound and reference systems used to estimate REs [77].

Another way to estimate the RE is to calculate it using quantum chemistry programs. When calculating the VRE, the higher energy structure has the same geometry as the optimal one (Figure 1) and the cyclic delocalization must be blocked. This approach removes all other effects and gives a more accurate determination of the RE, although the errors in this case come from the computational level. The accuracy of the calculation can be checked by comparing different calculated and experimental data. The block localized wavefunction (BLW) approach allows the estimation of the VRE by localizing certain bonds and disabling interactions between them [69,70,78,79]. Then, the energy difference between the delocalized and localized structures corresponds to VRE. The BLW method was implemented in Gamess in-house software [69,70,78,79], and is available in the Xiamen Valence Bond (XMVB) program, v. 2.0 [80], which should be incorporated in Gamess-US for BLW calculations. The Natural Bond Orbital (NBO) software [81,82] is implemented in or can be linked to the widely used Gaussian program. It provides a way to calculate VRE using the second-order perturbative estimates of donor–acceptor interactions in the NBO basis, which is part of the NBO output file, or using the NBO deletion (NBO_del) analysis [81,82,83]. Both BLW and NBO can optimize the structure with electronic interactions disabled. The energy difference between the resulting structure and the optimal delocalized one gives the adiabatic resonance energy (ARE, Figure 1), which is smaller due to the relaxation of geometry. The compression energy (CE) is the energy required to change the alternating bond lengths to the lengths of the delocalized structure [84], shown in Figure 1 for benzene. Figure 1 compares the VRE and ARE values obtained by BLW [69] and NBO_del [85] calculations. The BLW values are lower due to the optimization of the localized wavefunction, while in NBO calculations the occupied orbitals remain the same as they are optimized in the presence of deleted orbitals [86]. This difference between the methods should be kept in mind when using BLW and NBO, both of which are useful for assessing the energetic consequences of electron delocalization.

To get an idea of the impact of cyclic delocalization on energetic stabilization, we need to subtract the RE of an acyclic reference molecule from that calculated for the cyclic system. The resulting energy is called extra cyclic resonance energy (ECRE) [70]. A good reference molecule is one that has the same number of conjugations. For benzene, this is 1,3,5,7-octatetraene [70]. Figure 1 shows that ECREs based on the ARE of benzene are very similar when calculated with BLW and NBO methods, while NBO values based on VRE are larger.

The stabilization energy due to cyclic delocalization, also called aromatic stabilization energy (ASE), can also be estimated using energy decomposition analysis (EDA), which is a powerful method for providing insight into the nature of inter- and intramolecular interactions. Several EDA schemes have been developed, and common to all of them is the partitioning of the total interaction energy into physically meaningful components [87,88,89,90,91,92,93]. Thus, benzene was divided into the three open-shell (os) singlet fragments (Figure 2) and the total interaction energy between them (ΔE_int) was divided into the classical electrostatic attraction (ΔE_elstat), repulsive Pauli interaction (ΔE_Pauli), and stabilizing orbital interaction (ΔE_oi, Equation (1)). The latter energy term was further divided into contributions of σ (ΔE_σ) and π orbitals (ΔE_π). All-trans-1,3,5,7-octatetraene, decomposed into two doublets and two os singlets, served as a reference to calculate the ASE of 42.5 kcal/mol (Figure 2) [94].

ΔE_int = ΔE_elstat + ΔE_Pauli + ΔE_oi

(1)

ASE can also be calculated using the isodesmic [76,95] and homodesmotic reactions [76,96]. In the first, the type of bonds at both sides of the chemical equation are the same (single, double, triple, and CH), and bonds that are broken are the same as those which are formed. The homodesmotic reaction, which is a subgroup of isodesmic reactions, takes care that both sides of the equation have the same number of equally hybridized atoms, the same number of CC bond types (sp³-sp³, sp³-sp², sp²-sp², etc.), and the same number of CH bonds with equally hybridized carbon atoms. Figure 3 shows isodesmic (a) and homodesmotic reactions (b) used to calculate the ASE of benzene [97]. Schleyer and co-workers proposed the isomerization stabilization energy (ISE) to estimate the energy gain due to cyclic delocalization, which is the difference in energy between a system with disrupted cyclic delocalization and a system with delocalized electrons [98,99]. The developed methyl-methylene [98] and indene-isoindene [99] methods are shown in Figure 3c,d and can be also applied to study the aromaticity of the triplet state [100,101].

One more measure of energetic stabilization due to cyclic conjugation is a graph-theoretical approach solely based on the topology of the π-electron system. It was introduced independently by Aihara [102] and Gutman, Milun and Trinajstić [103], and is known as the topological resonance energy (TRE). The important feature of the method is the introduction of the acyclic polynomial which approximates the acyclic reference structure. TRE can be calculated for conjugated carbo- and heteromonocycles and polycycles, radicals and ions, Möbius annulenes, homoaromatic systems, and excited states [104,105].

4. Aromaticity and Electron Delocalization

The “aromatic” stabilization of molecules comes from electron delocalization. It cannot be measured experimentally, so a number of theoretical approaches to quantify electron delocalization and aromaticity have been developed. Only a few will be described here, while for a comprehensive review the reader is referred to reference [106].

Measures of Electron Delocalization

In the early 2000s, Solà and co-workers introduced the para delocalization index (PDI) [107,108]. It is based on the idea that delocalization of π electrons in benzene is greater between para- than between meta-related carbon atoms. PDI is calculated according to Equation (2) as the average of the three delocalization indices (δ) and can be used only for six-membered rings. The delocalization index quantifies the number of electrons which are delocalized or shared between the atoms and is derived from Bader’s Atoms in Molecules (AIM) theory [109,110]. PDI provides a measure of local aromaticity in polycyclic systems.

$$\mathrm{P}\mathrm{D}\mathrm{I}={\displaystyle \frac{\delta \left(\mathrm{1,4}\right)+\delta \left(\mathrm{2,5}\right)+\delta \left(\mathrm{3,6}\right)}{3}}$$

(2)

The aromatic fluctuation index (FLU) was developed somewhat later also by Solà and co-workers [108,111]. It provides information on the fluctuation of electronic charge between adjacent atoms in a ring and is given by Equation (3). The δ(A_i) is defined as in Equation (4) and α is a simple function to ensure that the first term is always greater than or equal to 1 (Equation (5)). A more delocalized ring has a lower FLU value, which is equal to zero in benzene. FLU can be calculated for rings of all sizes, individual rings in polycyclic compounds, or any other circuit. Calculating the FLU for all electrons requires reference values for δ, whereas they are not needed for calculating the delocalization of π electrons, called FLU_π, because δ_ref in Equation (3) is replaced with the average value of δ_π. Both PDI and FLU can be easily obtained using the Multiwfn software [112].

$$\mathrm{F}\mathrm{L}\mathrm{U}={\displaystyle \frac{1}{N}} \sum _{i=1}^N{\left[{\left({\displaystyle \frac{\delta \left(A_i\right)}{\delta \left(A_{i-1}\right)}}\right)}^{\alpha }\left({\displaystyle \frac{\delta \left(A_i,A_{i-1}\right)-{\delta }_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(A_i,A_{i-1}\right)}{{\delta }_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(A_i,A_{i-1}\right)}}\right)\right]}^2$$

(3)

$$\delta (A_i)=\sum _{A_j\ne A_j}\delta (A_i,A_j)$$

(4)

$$\alpha =\left\{\begin{array}{l}1 \delta \left(A_i\right)>\delta \left(A_{i-1}\right)\\ -1 \delta \left(A_i\right)\le \delta \left(A_{i-1}\right)\end{array}\right.$$

(5)

Electron density of delocalized bonds (EDDB) was developed by Szczepanik and co-workers [113,114]. It quantifies the number of delocalized electrons in a ring of any size or open path. The theory behind it and its mathematical definition can be found in refs. [113] and [114]. EDDB calculations can be performed using the runEDDB software [115], which needs files from NBO calculations.

In a recent study [116], the ability of PDI, FLU, and EDDB to track subtle changes in cyclic electron delocalization of the seven rings and perimeter of coronene that occur after the gradual replacement of its peripheral CC bonds by the isoelectronic BN bonds was examined (Figure 4). All three indices were found to agree well about the aromaticity of the outer benzene-like subunits, which are the most aromatic (blue color in the graphs). In the case of the central ring, the outlier point belongs to the ring of coronene, whose aromaticity is influenced by the six surrounding benzene-like rings (yellow color in the graphs). When it is excluded from the correlations, the R² values increase to 0.9989 (PDI/FLU), 0.9364 (EDDB/FLU), and 0.9230 (EDDB/PDI). The change in electron delocalization at the molecular perimeter is well described by FLU and EDDB (black color), but the reduced aromaticity of the individual BN rings, with small differences between them, is more difficult to describe (red color).

By studying the aromaticity of benzene, and mono- and polynitrobenzenes, Martínez-Araya with co-workers concluded that electron delocalization-based indices, such as PDI and FLU, are very sensitive to small variations in aromaticity [117]. In the original work that introduced FLU, calculations on the set of compounds shown in Figure 5 showed good agreement (r = −0.840) between PDI and FLU for six-membered rings [111].

When estimating aromaticity, it is useful to compare the degree of electron delocalization in the cyclic molecule and acyclic reference compound. Such a comparison of EDDB_π values was completed in a recent study of (anti)aromaticity of cyclo[2n]carbons [68], which are carbon allotropes that have two orthogonal π systems, shown in Figure 6 for cyclo[18]carbon. This compound was described as doubly aromatic based on 4n + 2 number of π electrons and development of diatropic ring currents under the influence of a perpendicular magnetic field [118,119,120,121]. However, the similar values of EDDB_π of C₁₈ and the linear C₁₈H₂ polyalkyne, together with similar RE per atom values for C₁₈ and C₂₀H₂, with the same number of conjugations as the cyclic molecule, and low ECRE values, led to the conclusion that C₁₈ is non-aromatic [68]. This conclusion was supported in a recently published paper by EDDB and EDLB (electron density of localized bonds) visualizations [122].

The example of C₁₈ shows that diatropicity alone is not sufficient to characterize a compound as aromatic. As will be discussed in Section 6, magnetic criteria for aromaticity must be supported by electronic and energetic ones.

5. Aromaticity and Molecular Geometry

One property of aromatic molecules is that they have (almost) equal bond lengths. Based on this, several structural indices of aromaticity have been developed [123,124]. A frequently used index is the Harmonic Oscillator Model of Aromaticity (HOMA) [125,126]. It is calculated according to Equation (6), where n is the number of bonds considered for summation, R_opt is the optimal bond length of the aromatic system, and R_i is the individual bond length of type j (CC, CN, CO, BN, etc.). Therefore, the HOMA of a completely aromatic system with all bonds equal to the optimal ones is 1. The normalization constant α is obtained so that the HOMA for a model non-aromatic compound is 0.

$$\mathrm{H}\mathrm{O}\mathrm{M}\mathrm{A}=1-{\displaystyle \frac{{\alpha }_j}{n}} \sum _i^n{(R_{\mathrm{o}\mathrm{p}\mathrm{t},j}-R_{j,i})}^2$$

(6)

Recently, two new parametrizations of HOMA were completed: HOMAc that includes antiaromaticity [127] and HOMER to be used for excited states [128].

Bond length equality is an important condition for efficient electron delocalization. Alternating bond lengths does not prevent delocalization, but reduces it and the resulting energetic stabilization. As mentioned above, the RE of the BLW-optimized 1,3,5-cyclohexatriene structure is 44% of the RE of benzene [70]. However, equal bond lengths are not necessarily a condition for efficient delocalization. Borazine (Figure 7) is isoelectronic to benzene and is known as “inorganic benzene.” It has all bonds equal and high HOMA = 0.940 [85] or even 1.000 [129], but is far less aromatic than benzene with ECRE and ASE of 22% [65] and 27% [94], respectively, of corresponding values for benzene. The reason for this reduced aromaticity is the large difference in electronegativity between boron (2.0) and nitrogen (3.0), which places the six π electrons mostly on the nitrogen atoms. Inserting a carbon atom between boron and nitrogen to form B-C-N units reduces the difference in electronegativity and makes carborazine (Figure 7) aromatic [130,131]. However, its HOMA = 0.812 and 0.771, calculated at the ωB97XD/def2tzvp and B3LYP/6-311+G(d,p) levels of theory, respectively, are lower than values of borazine. Similarly, the isoelectronic analogue of C₁₈, B₆C₆N₆ is more delocalized than B₉N₉ (Figure 7) [132].

Although the RE calculated for the planar structure with equal bond lengths of the heavier analogue of benzene, hexasilabenzene (Si₆H₆), is 53% of that of benzene [133], it prefers the nonplanar, chair-like structure (Figure 8) [134,135] due to the increased π-electron repulsion in the planar form [136]. Based on theoretical calculations, a cyclic delocalization of six electrons within the four-membered central ring has been proposed. It involves the lone pair, and 2 π and 2 σ electrons, and is called dismutational aromaticity [134,135].

The example of Si₆H₆ shows that even the π-system itself can have two opposite effects: stabilization due to delocalization and destabilization due to electron repulsion. Another example comes from the three isomeric azaborines (Figure 9). The charge separation of the π-electron system inherent in the 1,3-relationship of boron and nitrogen atoms destabilizes 1,3-isomer, but acts as a driving force for its strongest cyclic delocalization, as evidenced by its highest ECRE [65]. HOMA, PDI, and FLU are consistent with ECRE values [62].

Having considered the energetic, electronic, and structural criteria of aromaticity, let us once again analyze the IUPAC definition. If we interpret aromaticity as cyclic delocalization that makes a system thermodynamically more stable (relative to acyclic analogues) by overcoming all other (de)stabilizing effects, what do we call a cyclic delocalization that does not overcome other effects? As already mentioned, an easier way to define aromaticity is simply “a cyclic delocalization that stabilizes a delocalized electronic system (π or σ) more than an acyclic or interrupted cyclic delocalization does; it affects thermodynamic stability together with other stabilizing and destabilizing effects”. Another question is how to define an aromatic compound. One that is stabilized by aromaticity, or one which is experimentally viable because aromaticity prevails over (de)stabilizing effects? The latter might be more appropriate.

One can also ask which of the three mentioned indices should have priority when assessing aromaticity. As the case of borazine shows, they should be electronic and energetic, while equality of bond lengths is important for efficient delocalization, but is not always sufficient, as other factors may also play a role.

6. Aromaticity and Molecular Response to an External Magnetic Field

It is well known in nuclear magnetic resonance (NMR) spectroscopy that protons attached to an aromatic ring resonate at low field, while those above an aromatic ring resonate at high field [137,138,139]. The reason for this is that an external magnetic field acting at right angles to the plane of the ring induces a diamagnetic ring current in mobile π electrons, which generates a field that reinforces the external field at the area around the ring, but opposes it above/below the ring and in its central area. This is known as the ring current model (RCM), proposed by John A. Pople in 1956 [140]. NMR chemical shifts, reviewed by Mitchell [141], and magnetically induced ring currents, reviewed and discussed by Lazzeretti [142], Gomes and Mallion [143], Dickens and Mallion [144], Sundholm, Fliegl and Berger [145], Steiner et al. [146], and Fowler et al. [147,148], are widely used to study aromaticity and antiaromaticity. Anisotropy of the induced current density (AICD) was proposed by Herges and Geuenich for visualization of electron delocalization and ring currents [149,150].

In 1996, Schleyer and co-workers introduced nucleus-independent chemical shift (NICS) as a criterion of aromaticity [151,152]. NICS is negative for magnetic shielding (σ) and is calculated at the ring centers or a few angstroms above/below it to estimate tropicity, where negative NICS values are characteristic of diatropicity and positive values of paratropicity. Initially, the isotropic values (σ_iso), which are the average of the three magnetic shielding tensor components (σ_xx, σ_yy, and σ_zz), were used. Since ring currents are induced by a magnetic field acting normal to the plane of the ring, it was later shown that the out-of-plane (zz) component calculated 1 Å above the plane, NICS(1)_zz, and the π-electron contribution to the zz component calculated at ring centers, NICS(0)_πzz, are better measures of tropicity ((anti)aromaticity) [153]. Stanger performed an NICS scan in the direction perpendicular to the plane of the ring, starting from its center, to show that the resulting curves, corresponding to the out-of-plane component, have different shapes depending on the tropicity of the ring: for diatropic rings, a minimum is observed, and for paratropic, a maximum can be seen [154]. Tiznado and co-workers plotted one third of the out-of-plane component, −1/3(σ_zz), against one third of the sum of the two in-plane components, −1/3(σ_xx + σ_yy), calculated along a line perpendicular to the plane of the ring, starting from the center. The curves attain different shapes for diatropic rings (convex shape) and paratropic rings (concave shape), and a linear shape for non-aromatic systems. The point at which the contribution of the in-plane components is zero is called FiPC-NICS (free of in-plane components) [155]. NICS-rate, defined as ΔNICS/Δr, where r is the distance from the ring plane at which NICS is calculated starting from the center of the ring, was introduced by Noorizadeh and Dardab. The presence of a maximum or minimum in the NICS-rate versus r curve indicates diatropicity and paratropicity, respectively. When both the maximum and minimum are present, their absolute ratio (NICS-Rates Ratio) NRR = 0.5 is taken as the boundary between aromatic and nonaromatic rings [156]. Gershoni-Poranne and Stanger introduced NICS-xy-scan in which NICS_zz values are calculated along the x and y axes at the recommended height of 1.7 Å above the plane of the molecule [157]. Although the method does not calculate electron currents, it has been shown to be useful for identifying local, semi-global, and global ring currents in polycyclic systems, based on the magnetic shielding values. The conclusions are consistent with those based on current density maps [158]. Klod and Kleinpeter created 3D grids of points separated by 0.5 Å to calculate magnetic shielding in the region inside and around molecules. The resulting iso-chemical shielding surfaces (ICSSs) pictorially show anisotropic effects [159], which have different characteristics for aromatic, antiaromatic, and non-aromatic compounds. Figure 10 shows a deshielding region outside the benzene ring and a shielding region above/below and inside the ring. In the case of borazine, discontinuity in the deshielding region and a shorter shielding cone indicate its weaker aromaticity [160].

Karadakov and Horner used a denser 2D grid of 0.05 Å, providing more detailed information on chemical bonding and aromaticity [161]. A few examples are shown in Figure 11. It can be seen that the shielding maps, calculated 1 Å above the plane of the molecules, nicely visualize the local aromaticity corresponding to the rings having Clar’s sextets in phenanthrene (a) and triphenylene (b), and also the aromaticity originating from migrating Clar’s sextets in coronene (c). The discontinuity of the 15 ppm shielding in the rings of phenanthrene and triphenylene also implies the contribution of another resonance structure (Figure 11) [162].

Recently, Al-Yassiri developed tubular magnetic shielding scans (TMSSs) to study bonding and aromaticity based on the construction of cylindrical grids of points at which NICS values are calculated [163]. Islas, Heine, and Merino visualized the induced magnetic field (B_z^ind), corresponding to NICS_zz, to study tropicity of hydrocarbon and all-metal rings [164]. Muñoz-Castro and co-workers used the induced magnetic field visualizations to study spherical aromaticity of different structures, such as W@Au₁₂ and W@Si₁₆²⁺ clusters [165], boron clusters [166], and Au₂₀(PR₃)₈ [167]. Detailed reviews on NICS [168] and magnetic criteria of aromaticity [169,170] have been published.

Magnetically induced ring currents and the resulting magnetic shielding are very popular for assessing aromaticity and antiaromaticity. Many chemists make conclusions based on them, knowing that aromatic compounds develop diatropic ring currents, antiaromatic molecules paratropic ring currents, and non-aromatic systems generally have localized currents. Is this always the case? Foroutan-Nejad calculated magnetically induced ring currents of benzene radical cations and anions, which were strongly paratropic (MICD = −17.3, −47.4 and 12.0 nA/T for radical cation, anion, and neutral benzene, respectively). So, one could say that they are antiaromatic. However, both radical ions are energetically stabilized and have ISE = 7.4 and 6.7 kcal/mol, respectively (ISE = 33.5 kcal/mol for benzene). Radical anion of 1,2-diazine has higher ISE = 11.2 kcal/mol (ISE = 34.0 kcal/mol for neutral diazine), but also sustains strong paratropic ring current [171]. Are ring currents a good measure of (anti)aromaticity?

In 2001, Steiner and Fowler showed that the nature of the ring current is determined by the small number of electrons involved in occupied-to-unoccupied orbital transitions, that is HOMOs and LUMOs. A diatropic ring current originates when transition involves orbitals that differ in the number of nodal planes (n) by 1, and this is the case with aromatic systems as they have filled orbitals with the same n. In high symmetry structures of antiaromatic compounds, the degenerate HOMO level is half-filled. Such compounds reduce the symmetry to lift degeneracy. As a consequence, transition occurs between the orbitals with the same n leading to a paratropic current. The intensity of the current increases with the decreasing energy difference between occupied and unoccupied orbitals. Therefore, diatropicity and paratropicity arise from four and two HOMO electrons, respectively [172,173]. However, the energetic stabilization characteristic for aromatic systems includes all electrons that are cyclically delocalized. So, can we draw conclusions about aromaticity based on ring currents?

The following example will show that we must be careful when drawing such conclusions, as they may be coincidences. Coronene (Figure 12a), mentioned above, has six equally aromatic outer rings and a less aromatic central ring, which are conclusions based on electronic and structural criteria for aromaticity [116,174]. The NICS_πzz-x-scan (Figure 12b) is fully consistent with these conclusions. The two minima corresponding to NICS_πzz = −21.2 ppm indicate aromatic outer rings, while the maximum at NICS_πzz = −12.5 ppm corresponds to the weakly aromatic central ring. However, the ring currents of coronene consist of a peripheral diatropic current and a central paratropic current originating from the four HOMO electrons (Figure 12c–e) [172]. Therefore, the central ring is less shielded than the six outer rings, just as the NICS scan shows, because the two counter-rotating currents are the source of the shielding pattern. Here, we see that NICS results can explain aromaticity inferred from other measures even though such aromaticity is not the source of the NICS shielding pattern. So, chemists should be cautious when interpreting NICS. Two issues are important: keeping in mind Steiner’s and Fowler’s rules, and taking into account the electronic, energetic, and structural criteria of aromaticity, which must be used together with magnetic measures when assessing aromaticity.

Stabilization due to aromaticity is an intrinsic property of cyclic systems, while ring currents are a response to an external magnetic field. Therefore, it would be more accurate to speak of the tropicity of ring currents as a support to other indices whether the ring is aromatic or antiaromatic, but not to equate them with (anti)aromaticity.

Before moving on to the next section, it is worth mentioning an interesting prediction regarding diatropicity and paratropicity. In general, the reversal of diatropicity and paratropicity requires a change in electronic structure, such as oxidation, reduction, and excitation. Rončević and co-workers used theoretical calculations to show that molecular vibrations that control the change in the bond lengths of T₁ and S₁ cyclo[16]carbon cause a sudden change in the nature of ring currents, where bond length alternating the D_8h structure is diatropic, while the D_16h structure with BLA = 0 is paratropic. The explanation lies in the fact that C₁₆ has two orthogonal π-electron systems which are diatropic and paratropic in T₁ and S₁ electronic states. The Kekulé vibrations change the relative strength of the paratropic and diatropic currents leading to the reversal of the nature of overall ring currents [175].

7. Antiaromaticity

The term “antiaromaticity” was first used by Breslow to describe a compound in which cyclic delocalization is destabilizing relative to an open-chain conjugated system. Breslow with co-workers investigated the base-catalyzed exchange of hydrogen with deuterium in compounds 1–4 (Figure 13). They found a much slower exchange rate in unsaturated 1 and 3 compared to the saturated 2 and 4, explaining it as a conjugative destabilization or antiaromaticity [176].

Antiaromatic compounds have 4n π electrons in a ring that are mostly localized. According to the IUPAC definition, “Those cyclic molecules for which cyclic electron delocalization provides for the reduction (in some cases, loss) of thermodynamic stability compared to acyclic structural analogues are classified as antiaromatic species” [177]. In addition, it is stated that “antiaromatic molecules possess negative (or very low positive) values of resonance energy” [177]. However, the resonance energy cannot be negative. Only ECRE or ASE can. For example, ECRE = −10.5 kcal/mol is calculated for the antiaromatic paradigm, cyclobutadiene (CBD), using the BLW method and all-trans 1,3,5-hexatriene as a reference [70]. Antiaromatic compounds are generally difficult to synthesize because they are not thermodynamically stable. Does cyclic electron delocalization destabilize such compounds?

First, these compounds are mostly localized. Second, electron delocalization can only have a stabilizing effect. What, then, makes antiaromatic molecules unstable? Let us analyze the stability of CBD. This molecule has never been isolated, but characterized when trapped in a matrix [178]. Its ground state is a rectangular singlet. The VRE of cyclobutadiene calculated by the BLW method is 10.9 kcal/mol [70], which means that it is stabilized by electron delocalization, while angular strain, torsional strain, repulsion between parallel π bonds, and the resulting C-C bond elongation destabilize CBD by 62.2 kcal/mol [179].

Another example is cyclooctatetraene (COT). It adopts a nonplanar tub-shaped conformation with D_2d symmetry, which is often interpreted as the molecule’s tendency to mitigate antiaromaticity. What does it exactly mean? In the planar conformation, the molecule experiences significant torsional strain because all CC and CH bonds are eclipsed. In its actual conformation, determined by the low-temperature X-ray analysis, the CCCC dihedral angles around the four bonds increase to 55–58° approaching the ideal value of 60°, and HCCH dihedral angles increase to 41–46° [180]. The CCC bond angles of ~126°, determined by X-ray analysis [180] and electron diffraction [181], also approach the ideal value of 120°, while those in the planar octagon form are much larger, 135°. Therefore, the molecule is relieved of torsional and angular strain by adopting a non-planar conformation, while the conjugative π → π* stabilization is reduced by 33.6 kcal/mol, as estimated by NBO_del analysis [182]. Also, the tub-shaped form is stabilized by hyperconjugation [182].

Therefore, a better definition of antiaromaticity can be “little or no stabilization due to cyclic conjugation in strained systems.” It can be assessed using the aforementioned aromaticity criteria. Antiaromatic compounds have small or negative HOMA and HOMAc values, small or negative ECRE and ISE, small values of PDI and EDDB, and high FLU values. However, non-aromatic compounds can also have similar values of structural, energetic, and electronic indices as antiaromatic ones and one may wonder how to distinguish between non-aromatic and antiaromatic compounds. During a roundtable discussion at the Aromaticity 2025 conference (Merida, Mexico, 27–30 January), Muñoz-Castro proposed the nature of ring currents as a way to make the distinction: antiaromatic rings develop paratropic ring currents, while non-aromatic rings generally have localized currents. However, it is not always easy to make the distinction solely on the basis of the magnetic criterion of (anti)aromaticity. For example, cyclo[8]carbon and cyclo[16]carbon, with two orthogonal π_in and π_out electron systems, both exhibit a shielding pattern characteristic of paratropic ring currents induced by an external perpendicular magnetic field [118,119]. The shielding cone of the latter is longer-ranged (Figure 14a) [119], but the smaller deshielding values were calculated on a 2D grid placed 1 Å above the ring plane (Figure 14b) [118]. Separating the total shielding into contributions from the two π systems showed that the π_in system of C₈ is very weakly paratropic due to the large bond angle alternation (BAA), while both π systems contribute to the shielding cone of C₁₆ (Figure 14a) [119]. The ring current strength of C₁₆ (−24.3 nA/T) is weaker than that of C₈ (−35.7 nA/T), and both currents are paratropic [183]. AICD shows strong paratropicity of π_out of C₈ and much weaker for both systems of C₁₆ with a tendency towards localization (Figure 14c) [184]. The total shielding calculated at the center of C₈ and C₁₆ is σ_in/σ_out = −33.1/−105.9 ppm and −24.2/−20.1 ppm, respectively [184]. Thus, the π_out of C₈ is much more paratropic.

A detailed analysis of structural, electronic, and energetic measures of (anti)aromaticity characterized C₁₆ and larger cyclo[2n]carbons as non-aromatic [68]. Considering that the π_out of C₈ and both π systems of C₁₆ are mostly localized with small negative ECRE = −11.7 kcal/mol and −10.9 kcal/mol, respectively, strong paratropicity, quantitatively assessed as ring current strength and specific NICS values, may be a point of distinction between antiaromaticity and non-aromaticity. However, as the previous discussion shows, magnetic criteria of (anti)aromaticity must be supported by electronic and energetic measures. As a note, the π_in of C₈ is described as moderately stabilized by weak delocalization [68].

C₁₆ was synthesized [185], but C₈ has not yet been synthesized, probably because it is more strained than C₁₆ [68]. Thus, an antiaromatic compound can be defined as “one that is difficult to synthesize because it is strained and poorly stabilized by cyclic delocalization and that exhibits strong paratropicity under the influence of an external magnetic field acting at right angles to the plane of the ring.”

8. Aromaticity and Antiaromaticity in Excited States: Baird’s Rule

Hückel’s 4n + 2 aromaticity rule applies to the closed-shell ground state. In 1972, Baird proposed that the π-electron counting rules are reversed for the first ππ* excited triplet state, which means that the 4n + 2 π-electron species become antiaromatic and those with 4n π electrons become aromatic upon excitation [186]. Later theoretical studies supported the proposal, now known as Baird’s rule, for both carbocycles and heterocycles. They are based on studies of ASE [187,188] and ISE [100,101], π-electron delocalization [189], π-electron contribution to the electron localization function [190], molecular structure [187,191], magnetic shielding [187,188], and induced ring currents [192]. In 2008, Karadakov showed that the rule also applies to the first excited singlet state, studying the magnetic shielding of S₁ benzene [193], square cyclobutadiene [193], and D_8h cyclooctatetraene (COT) [194]. However, the second excited singlet states of benzene and cyclobutadiene were characterized as aromatic and antiaromatic, respectively [195]. Baird’s rule has also been experimentally confirmed by spectroscopic measurements of hexaphyrins [25], by studying the inversion barriers of chiral thiophene-fused cyclooctatetraenes [196] and metal-free photochemical silylation of antiaromatic benzene, while aromatic COT was unreactive (Figure 15) [197].

The concept of excited-state aromaticity (ESA) and excited-state antiaromaticity (ESAA) was used to show that the light-driven proton transfer reactions happen due to ESAA relief [198,199]. For example, ortho-salicylic acid is known to undergo intramolecular proton transfer in its S₁ state, but not in the S₀ and S₂ states. This is because the proton transfer in the S₁ state reduces antiaromaticity, while the S₀ and S₂ states are both aromatic (Figure 16) [198].

The Watson–Crick base pairs adenine–thymine and guanine–cytosine were able to encode genetic information long before the ozone layer formed. How did they survive the harsh conditions of UV radiation? According to the work of Karas, Wu, Ottosson, and Wu, upon UV irradiation, the purine base becomes highly antiaromatic. To alleviate this antiaromaticity, the base pair undergoes a very rapid electron and proton transfer to achieve an aromatic charge-transfer state, and then returns to its initial state via non-radiative decay (Figure 17, shown for the guanine–cytosine base pair). This process is much faster than any other reaction that can damage the bases [200].

In a computational work, Wang, Oruganti, and Durbeej showed that the efficiency of molecular motors based on E/Z isomerization can be improved by designing a system that is capable of donating an electron in its photoactive state from one part (ring) to the other ring, which then becomes an aromatic anion (Figure 18) [201,202].

Ottosson and co-workers published comprehensive review articles that include computational and experimental evidence for ESA and ESAA, and rationalize the previously reported photochemical and photophysical behavior of molecules and novel reactions in light of these two concepts [22,26,27]. In a recent review article, Dos Santos, Wu, and Alabugin discuss how ESA and ESAA affect the photocyclization reactions of alkenes and arenes, many of which lead to the formation of complex molecular systems [203].

9. Is (Anti)aromaticity Characteristic of Small Rings?

Matito with co-workers studied the (anti)aromaticity of neutral [n]annulenes with an even number of atoms (n = 4–18), in their singlet ground state and the first excited triplet state. They found that the molecules lose antiaromaticity faster than aromaticity with an increasing number of atoms [204]. Based on structural, electronic, energetic, and magnetic criteria for (anti)aromaticity, Van Nyvel, Alonso, and Solà also concluded that (anti)aromaticity decreases with increasing ring size, in neutral and charged annulenes in singlet and triplet states. They found that the ASE for large rings is small [205]. In addition, it was also found that the (anti)aromaticity in the excited state is weaker than the (anti)aromaticity in the singlet ground state [205]. Similar conclusions were obtained by studying the sensitivity of the triplet state aromaticity of cyclopentadienyl cations to substituent effects, which is more sensitive than the ground state aromaticity of benzene [206]. By calculating the strength of magnetically induced ring currents in cyclo[n]carbons with an even number of atoms (n = 6–100), Baryshnikov, Valiev, and co-workers found that Hückel rules for aromaticity and antiaromaticity become degenerate for n > 50, when the molecules begin to have localized currents [183]. Ottosson and co-workers studied the triplet state aromaticity of macrocycles composed of different monocycles. They found a competition between the macrocyclic aromaticity and closed-shell aromaticity of the individual rings, concluding that monomers with smaller Hückel aromaticity are a better choice for achieving triplet state aromaticity of macrocycles [207]. The first experimental evidence for ASE in large macrocycles, π-conjugated porphyrin nanorings, was obtained by Anderson and co-workers by measuring the oxidation potential and the barrier to conformational change. The authors concluded that the effect of aromaticity on the stability of large systems is small, but measurable [208].

Aromaticity, as a stabilizing effect due to cyclic conjugation, seems to be a privilege of small rings. If we say that antiaromaticity is weak or has no stabilization due to cyclic conjugation, its relationship to small rings seems to arise from the destabilization of small rings by strain. Cyclic delocalization in large rings is more similar to delocalization in acyclic systems.

10. Conclusions

After reviewing and discussing the selected literature on aromaticity and antiaromaticity, the following can be concluded about these two concepts.

Both are related to the cyclic arrangement of electrons. In the case of aromaticity (4n + 2 number of electrons), the electrons are delocalized and such delocalization stabilizes the delocalized electronic system more than is the case in an acyclic system or a system with a disrupted cyclic arrangement (or conjugation in the case of π electrons). Aromaticity affects thermodynamic stability along with other (de)stabilizing effects. Delocalization is more efficient when bond lengths are similar, so the structural criterion of aromaticity is important, in addition to delocalization and energetic stabilization.

In the case of antiaromaticity (4n number of electrons), the electrons are mostly localized, so that the small delocalization, if any, stabilizes that electronic system less than in an acyclic molecule or a molecule with a disrupted cyclic arrangement of electrons. Molecules that we call antiaromatic are destabilized by strain, which, combined with the small stabilization due to cyclic (de)localization, makes such molecules difficult to synthesize.

The most popular way to distinguish between aromaticity and antiaromaticity is the nature of ring currents induced by an external magnetic field. However, ring currents are a molecular response to an external stimulus and their nature is determined by the small number of high-energy electrons. Therefore, it is suggested that conclusions based on the nature of ring currents should be supported by electronic, energetic, and structural measures of (anti)aromaticity. Also, diatropicity and paratropicity should not be equated with aromaticity and antiaromaticity because (anti)aromatic stabilization is an energetic phenomenon that occurs by cyclic electron delocalization when the molecule is free from external influences.

There is an important question of how to distinguish between antiaromaticity and non-aromaticity as both are characterized by weak energetic stabilization due to weak delocalization. Since antiaromaticity is characteristic of small rings, the distinction can be made on the basis of strong paratropicity occurring in antiaromatic compounds, but mostly localized currents and weak paratropicity in non-aromatic systems. “Strong” and “weak” paratropicity can be distinguished by the ring current strength and specific NICS values.

The concepts of excited-state aromaticity and antiaromaticity have proven to be very useful for explaining and predicting the photochemical and photophysical behavior of molecules, including those of biological importance, for planning the synthesis of simple and complex molecular structures, and for designing molecular machines.