Linearly Expanded Azulenes in the ${\text{S}}_0$, ${\text{T}}_1$, and ${\text{Qu}}_1$ States: A Unified View of Electronic Structure and Aromaticity

ABSTRACT

Linearly expanded azulenes (LEAs) are isomers of polyacenes (PAs) in which the two terminal benzene rings are replaced by a five‐ and seven‐membered ring, respectively. Despite this seemingly minor structural perturbation, LEAs exhibit electronic behavior fundamentally distinct from their PA counterparts. Herein, we systematically investigate the electronic structure and magnetic aromaticity of a series of LEAs in their ${\text{S}}_0$, ${\text{T}}_1$, and ${\text{Qu}}_1$ states. We find that LEAs begin to exhibit biradical character at smaller molecular sizes than the corresponding PAs, driven by preferential localization of unpaired electrons on the terminal rings, which allows for the formation of a migrating Clar sextet in the central benzenoid chain and results in a distinctive pattern of paratropic terminal domains and diatropic central domains. This same preference causes the two families to show opposite locations of unpaired electrons in ${\text{T}}_1$. In ${\text{Qu}}_1$, it results in a superposition of the two ${\text{T}}_1$‐state patterns, whereby LEAs localize unpaired electrons at the terminal rings while the benzenoid core adopts the PA‐like arrangement. Together, these results deepen our understanding of non‐benzenoid open‐shell molecules and underscore the critical role of ring topology in governing electronic structure and aromatic character.

Linear expansion of azulene leads to markedly different electronic behavior from that of the corresponding acenes, including an earlier emergence of open‐shell character. Simple resonance structures provide an intuitive framework for understanding these trends and their evolution with molecular size.

1. Introduction

Polycyclic aromatic hydrocarbons (PAHs) continue to occupy a central position in organic chemistry, serving both as model systems to understand fundamental aspects of electronic structure and reactivity [1], and as functional materials for applications in organic electronics [2, 3, 4]. A defining feature of these $\pi$‐conjugated systems is their sensitivity to molecular topology: seemingly minor changes in ring size, connectivity, or annelation can profoundly alter electronic structure [5, 6], aromaticity [7, 8, 9], optical response [10], and physicochemical properties [11].

One of the best known examples is the naphthalene‐azulene isomer pair (Figure 1). Naphthalene comprises two fused six‐membered rings (6MRs), whereas azulene is composed of fused five‐ and seven‐membered rings (5MR and 7MR, respectively). Despite sharing the same molecular formula and 10$\pi$‐electron framework, the two molecules exhibit remarkably different properties: Naphthalene is colorless, essentially nonpolar, and has conventional fluorescence, while azulene is named for its distinct blue color, has a large dipole moment (1.08 D), and famously violates Kasha's rule by fluorescing from its ${\text{S}}_2$ state [12, 13].

FIGURE 1.

Polyacene and linearly expanded azulene isomers with notation used in this text is detailed. The 6MRs corresponding to the naming practice are outlined in bold.

Consequently, azulene‐containing PAHs have attracted sustained interest for decades [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], with renewed attention driven by their potential for (opto)electronic applications [31, 32, 33, 34, 35, 36, 37, 38]. While these studies demonstrate that the azulene motif provides a powerful means of tuning the properties of extended PAHs, they cannot answer the more fundamental question: How do the dramatic consequences of the naphthalene–azulene isomerization evolve as the conjugated framework is systematically extended?

The natural extension of the naphthalene–azulene comparison is the homologous series of polyacenes (PAs) and their constitutional isomers, linearly expanded azulenes (LEAs), obtained by replacing the terminal 6MRs with a 5MR and a 7MR, respectively. While PAs have long served as benchmark systems for understanding the evolution of electronic structure with conjugation length [3, 39, 40], comparatively little is known about LEAs. To date, the parent LEAs have not been synthesized, and only a few molecules containing the corresponding 5–6–7 motif have been reported [41, 42, 43].

Computationally, LEAs have been examined in several previous studies, although primarily from the perspective of their electronic structure, spin state, and optical properties. Aihara predicted as early as 1975 that the 5‐6‐6‐6‐7 structures (our Az ^3B) should have a triplet ground state [44], but it is only in more recent years that modern computational methods have been used to validate this prediction. Radenković and coworkers used broken‐symmetry DFT and CASSCF methods to examine their ground‐ and first triplet‐state electronic structures [45], while Champagne and coworkers investigated the open‐shell (OS) character, singlet–triplet energetics, and nonlinear optical response of Az ^3B using spin‐flip time‐dependent DFT [46]. More recently, Ghosh and coworkers examined the same molecular family using full‐valence DMRG and other multireference approaches [47]. Collectively, these studies established that LEAs develop pronounced OS character at shorter lengths than the corresponding PAs and that their singlet–triplet gaps decrease rapidly with increasing molecular size. Radenković also showed that the triplet‐state spin density localizes on the terminal odd‐membered rings.

Because aromatic stabilization, spin localization, and ring currents are intimately coupled in OS $\pi$‐systems, magnetic aromaticity provides a natural framework for exploring the impact of azulene topology. In particular, we aim to elucidate how the azulene topology reshapes magnetic (anti)aromaticity across different spin states, how this is related to the spatial distribution of unpaired electrons, and how LEAs compare to PAs with respect to these properties. Moreover, characterization of the first quintet state, ${\text{Qu}}_1$ is particularly compelling in light of recent work by Ottosson and coworkers [13, 48], who demonstrated that the ${\text{S}}_2$ state of azulene shares a closely analogous electronic structure with its ${\text{Qu}}_1$ state.

In the present work, we address these questions through a systematic computational investigation of the electronic structure and aromaticity of a series of LEAs, including direct comparison to their isomeric PAs (Figure 1).

2. Computational Methods

All geometry optimizations were performed using the Gaussian 16 suite of programs, revision C.01 [49]. Geometry optimizations for all molecules in the singlet state were performed with the CAM‐B3LYP [50] functional and the def2‐TZVP basis set [51], following a thorough benchmarking process (see Section S2 of the Supporting Information for additional details). All calculations were carried out within the unrestricted Kohn–Sham (UKS) formalism. All wavefunctions were tested for stability using the stable=opt keyword. Harmonic frequency calculations were carried out at the same level of theory to confirm that all reported structures are true minima on the potential energy surface, that is, $N_{imag}=0$.

The OS character of the longer open‐shell singlet (OSS) LEAs and PAs was evaluated using bi/diradical ¹ character indices $⟨S^2⟩$ value and Yamaguchi's index $y_0$ [54], calculated with both unrestricted Hartree–Fock (UHF) and unrestricted Kohn–Sham (UKS) formalisms (see Section S2 of the Supporting Information for further details and discussion).

The magnetic aromaticity [55] of the molecules was evaluated using current density [56] and nucleus‐independent chemical shift (NICS) [57, 58, 59, 60]. $\text{NICS}{(1)}_{\pi zz}$ values were computed with the gauge‐including atomic orbitals (GIAO) method [61, 62, 63, 64], with input generation via Aroma [65], and dissection to $\pi$‐contributions using Stanger's $\sigma$‐only model [66]. Current density plots for the $\pi$‐currents were obtained with the SYSMOIC package [67] using the continuous set of gauge transformations (CSGT) method [68] (see Section S1 of the Supporting Information for the list of $\pi$‐orbitals).

We note that the OSS states of Az ^2B—Az ^4B possess pronounced multireference character, for which multiconfigurational methods would provide a more rigorous treatment. However, full geometry optimization and harmonic frequency calculations at such levels of theory are computationally intractable for systems of this size and across the three spin multiplicities investigated here. Importantly, previous higher‐level studies on the same LEA series, including CASSCF, full‐valence DMRG, and spin‐flip time‐dependent DFT calculations, all support the qualitative electronic‐structure picture obtained here, particularly the emergence of pronounced OS character in the longer LEAs and a quasi‐degeneracy of singlet and triplet states [45, 46, 47]. The quantitative singlet–triplet energetics, which are more method‐dependent, are discussed explicitly below.

Comparison to Radenković and coworkers' work revealed an even closer agreement between our CAM‐B3LYP results and their CASSCF results than their reported agreement with B3LYP, lending additional confidence to the functional choice. Importantly, their multiconfigurational spin density calculations show that the unpaired electrons localize at the terminal odd‐membered rings. Because the primary aim of the present work is the characterization of magnetic aromaticity across spin states rather than the precise determination of ${\text{S}}_0$‐${\text{T}}_1$ energy gaps, we concluded DFT calculations with the UKS formalism provide sufficient grounds for the conclusions drawn here.

3. Results and Discussion

All molecules are studied in three electronic states: ${\text{S}}_0$, ${\text{T}}_1$, and ${\text{Qu}}_1$. We denote the spin state with the respective preceding superscript, for example, ¹ Np ^0B is naphthalene in the singlet state. Additionally, for conciseness, we use the term ${\text{S}}_0$, even when the lowest singlet state was found to be OS.

This section is arranged as follows: We begin by analyzing the energetic differences between the LEA and PA isomers in each state, then present and discuss the magnetic (anti)aromaticity in each state and its relation to spin density distribution. Finally, we provide a rationalization for these observations within a unified conceptual framework based on resonance structures (RSs) and graph‐theoretical concepts, including Clar's rule [69, 70], Randic's conjugated circuit method [71, 72, 73], and Mandado's analysis for OS systems [74].

3.1. Energetic Comparison

Figure 2A compares the relative energies of the two families across the three spin states. In all cases, the PAs are thermodynamically more stable than the corresponding LEAs, although the energy difference decreases with increasing multiplicity.

FIGURE 2.

(A) Relative energies between isomer pairs of PAs and LEAs in the three spin states. (B) $\mathrm{\Delta }{E}_{\mathrm{S}-\mathrm{T}}$ for PAs (purple) and LEAs (teal).

In ${\text{S}}_0$, with the exception of the parent ¹ Az ^0B and ¹ Np ^0B pair (38 kcal/mol difference), all other pairs are ∼50–55 kcal/mol apart and the gaps decrease slightly along the series. This nearly constant destabilization is comparable to the strength of a C=C $\pi$‐bond ($\sim$65 kcal/mol), suggesting that the isomerization imposes a nearly size‐independent penalty on the $\pi$‐system. In contrast, in both ${\text{T}}_1$ and ${\text{Qu}}_1$, the relative instability of the LEA isomer grows with size, from $\sim$10 to $\sim$30 kcal/mol in ${\text{T}}_1$ and from $\sim$10 to $\sim$20 kcal/mol in ${\text{Qu}}_1$. Considering that most LEAs have quasi‐degenerate ${\text{S}}_0$ and ${\text{T}}_1$ states, this suggests that chain elongation destabilizes the ${\text{T}}_1$‐PAs less than the ${\text{S}}_0$‐PAs.

Figure 2B compares the $\mathrm{\Delta }{E}_{\mathrm{S}-\mathrm{T}}$ for LEAs and PAs as a function of molecular size. For PAs, ${\text{S}}_0$ is always markedly lower than ${\text{T}}_1$, with $\mathrm{\Delta }{E}_{\mathrm{S}-\mathrm{T}}$ decreasing monotonically from 64 (Np ^0B) to 13 kcal/mol (Np ^4B), consistent with the formation of diradical character in the singlet state of the latter ($y_0^{\text{UHF}}$ = 0.568, $⟨S^2⟩$ = 0.961), and in agreement with the results of Bendikov [40], Hachmann [75], and others [76, 77].

The LEA series shows a much faster collapse. While Az ^0B and Az ^1B maintain well‐separated singlet ground states, Az ^2B, Az ^3B, and Az ^4B have essentially isoenergetic ${\text{S}}_0$ and ${\text{T}}_1$ states at the CAM‐B3LYP level ($\mathrm{\Delta }{E}_{\mathrm{S}-\mathrm{T}}$=0.3, 0.6, and $-2.1$ kcal/mol, respectively), and show marked OS character (Az ^2B: $y_0^{\text{UHF}}$ = 0.798, $⟨S^2⟩=0.969$; Az ^3B: $y_0^{\text{UHF}}$ = 0.830, $⟨S^2⟩$ = 1.100; Az ^4B: $y_0^{\text{UHF}}$ = 0.860, $⟨S^2⟩$=1.18; see Section S2 in the Supporting Information)). The precise $\mathrm{\Delta }{E}_{\mathrm{S}-\mathrm{T}}$ values are method‐dependent. Radenković and coworkers' CASSCF calculations predict $\mathrm{\Delta }{E}_{\mathrm{S}-\mathrm{T}}$=8.3, 6.4, and 4.3 kcal/mol, respectively [45], whereas the full‐valence DMRG calculations of Ghosh and coworkers on the same series predict values of 10.7, 0.9, and 0.35 kcal/mol, respectively [47]. Thus, the different methods do not agree quantitatively, and the small negative CAM‐B3LYP value for Az ^4B should be interpreted as evidence of a small singlet–triplet separation rather than as a firm prediction of a triplet ground state. Regardless, the general trend is clear: $\mathrm{\Delta }{E}_{\mathrm{S}-\mathrm{T}}$ in LEAs collapses dramatically with molecular size, reaching a low‐gap regime well before the isomeric PA family. Importantly, our analysis below concerns the electronic structure and magnetic response within each individual spin state and therefore does not depend on the precise energetic ordering of ${\text{S}}_0$ and ${\text{T}}_1$. Moreover, despite these differences, all methods agree on the critical issue of electron placement; they all predict disjoint biradical character, in which the two unpaired electrons are spatially confined to the terminal odd‐membered rings with minimal exchange interaction.

3.2. Magnetic Aromaticity

Magnetic aromaticity is analyzed through a combination of current density plots and $\text{NICS}{(1)}_{\pi zz}$ values, which provide both a pictorial depiction of the currents in the molecules, and a quantitative metric that allows for comparison among them. For OS cases, in which Baird aromaticity [78, 79] becomes relevant, spin density plots are depicted.

3.2.1. The ${\text{S}}_0$ State

The ground‐state electronic structure and magnetic aromaticity of PAs are well established [40, 80, 81]. We therefore summarize only the main trends to provide a reference for comparison with the LEAs. Current density plots (Figure 3) show the expected global diatropic current around the symmetric ¹ Np ^0B and additional local circulations in the subsequent closed‐shell (CS) isomers, ¹ Np ^1B–¹ Np ^3B, consistent with their symmetry—that is, around the central ring in ¹ Np ^1B, around the two central rings in ¹ Np ^2B, and around both the central ring and central tricyclic motif of ¹ Np ^3B. The superposition of these currents gives rise to the observed $\text{NICS}{(1)}_{\pi zz}$ values, which grow increasingly negative towards the central rings as the PA elongates ($-39$, $-39$, and $-42$ ppm, in the central rings of ¹ Np ^1B, ¹ Np ^2B, and ¹ Np ^3B, respectively). This trend shifts with ¹ Np ^4B, which is the first PA to have an OSS ground state ($y_0^{\text{UHF}}$ = 0.568). For ¹ Np ^4B, a global current is still observed, but the strong center‐based currents are replaced by weaker ones localized on the second and fifth (penultimate) rings, which also show the most negative $\text{NICS}{(1)}_{\pi zz}$ values in the molecule ($-29$ ppm).

FIGURE 3.

Magnetic aromaticity and spin density plots of ${\text{S}}_0$‐PAs and ${\text{S}}_0$‐LEAs. (A) Current density plots depicted 1 Å above the molecular plane. (B) $\text{NICS}{\left(1\right)}_{\pi zz}$ values (ppm). (C) Spin density plots depicted at isosurface value $\alpha$=0.02. The rings with the greatest spin density are highlighted in pink. Double bond omitted for clarity.

Turning to the LEAs, ¹ Az ^0B shows relatively similar behavior to ¹ Np ^0B, as both the 5MR and 7MR are strongly diatropic (5MR: $-47$ ppm; 7MR: $-26$ ppm, Figure 3). In contrast to ¹ Np ^0B, however, the current density plot reveals that this stems from a superposition of a global diatropic current and a local current in the 5MR. We note that the observation of a local current in ¹ Az ^0B contradicts the report of Ottosson and coworkers, which claimed no evidence exists for transannular delocalization in ¹ Az ^0B[13], but is in agreement with the DFT currents reported by Pino Rios [82] and the topological currents reported by Kerim [83], who similarly found local currents. See Section S3 of the Supporting Information for additional analysis.

Insertion of a single 6MR between the two rings, to form Az ^1B, impacts the 7MR much more strongly than the 5MR ($-8$ and $-43$ ppm; differences of $+18$ and $+4$ ppm, respectively), and the current density plot shows three separate circuits: A local diatropic 5MR current, a semi‐global 5MR–6MR current, and a weaker global current around the periphery of the entire molecule.

Dramatic changes are observed upon insertion of additional 6MRs to form ¹ Az ^2B, ¹ Az ^3B, and ¹ Az ^4B. For these molecules, both the 5MR and 7MR display paratropic currents, which are noticeably stronger in the 7MR. In addition, there are a semi‐global diatropic current encompassing the 5MR and all 6MRs and a series of PA‐type currents in the 6MR sections of the molecules. Altogether, the superposition of currents produces paratropic 7MRs (22, 15, and 13 ppm), essentially atropic 5MRs (3, 3, and 2 ppm), and PA‐like diatropic 6MRs.

Notably, as with the PAs, the marked shift in behavior coincides with the transition from CS to OS ground states. Aromaticity in OSS systems is commonly interpreted through Baird's rule [78, 79], which predicts that Hückel aromaticity reverses upon excitation. In polycyclic molecules, however, aromaticity reverses only in the region bearing the unpaired electrons [6, 84, 85], while regions devoid of spin density retain essentially ${\text{S}}_0$‐like behavior. Figure 3C illustrates this directly: In both families, the paratropic/less diatropic domains are the rings bearing the most spin density—the central 6MRs in the PAs, and the terminal odd‐membered rings in the LEAs.

3.2.2. The ${\text{T}}_1$ State

As for the ${\text{S}}_0$ state, we summarize only the established ${\text{T}}_1$ behavior of the PA series [85, 86, 87, 88, 89] to provide a reference for comparison with the LEAs. The bicyclic ³ Np ^0B exhibits the expected global paratropic ring current (24 ppm). Incorporation of additional 6MRs results in loss of this global current and the formation of localized currents of opposite character: the termini develop distinct diatropic currents while the central rings retain paratropic‐to‐weakly‐diatropic character (Figure 4). Both regions show the same length dependency. The longer the PA, the less paratropic its central region (³ Np ^1B: 23; ³ Np ^4B: $-2$ ppm) and the more diatropic its terminal regions become (³ Np ^1B: $-8$; ³ Np ^4B: $-28$ ppm).

FIGURE 4.

Magnetic aromaticity and spin density plots of ${\text{T}}_1$‐PAs and ${\text{T}}_1$‐LEAs. (A) Current density plots depicted at 1 Å above the molecular plane. (B) $\text{NICS}{(1)}_{\pi zz}$values (ppm). (C) Spin density plots depicted at the isosurface value $\alpha =0.02$. The rings with the greatest spin density are highlighted in pink. Double bonds omitted for clarity.

Like the PAs, the bicyclic 10$\pi$e ³ Az ^0B also displays a global paratropic ring current, which resolves into local circuits upon addition of 6MRs. In contrast to the PAs, however, this localization produces the opposite polarity: The terminal odd‐membered rings remain paratropic while the central benzenoid region becomes diatropic. Despite the reversed polarity, the evolution with molecular size is remarkably similar in the two families. As the LEAs elongate, the central benzenoid region becomes progressively more diatropic (³ Az ^1B: $-6$ ppm; ³ Az ^4B: $-32$ and $-30$ ppm), while the terminal odd‐membered rings become progressively less paratropic (³ Az ^1B: 19 and 20 ppm; ³ Az ^4B: 3 and 10 ppm). Thus, the topology reverses the spatial distribution of magnetic aromaticity without altering its size dependence.

As in the OSS states, the ${\text{T}}_1$ magnetic response is also viewed through the lens of Baird's rule. Figure 4C confirms that, for both families, the paratropic domains are those hosting the greatest spin density and, like in ${\text{S}}_0$, these regions are the central 6MRs in the PAs and the terminal odd‐membered rings in the LEAs.

3.2.3. The ${\text{Qu}}_1$ State

Characterization of PAs and LEAs in their ${\text{Qu}}_1$ states is not necessarily inherently interesting, as this state is unlikely to be accessed under most reaction conditions. Rather, ${\text{Qu}}_1$ serves as a computationally accessible proxy for the ${\text{S}}_2$ state. Recently, Ottosson, Slanina and coworkers demonstrated that the well‐known anti‐Kasha behavior of azulene can be explained by the contrasting aromatic and antiaromatic characters of its ${\text{S}}_2$ and ${\text{S}}_1$ states. Furthermore, they showed that the electronic structure of the ${\text{S}}_2$ state is analogous to the ${\text{Qu}}_1$ state [13]. Assuming this relationship holds for the other members of the family, ${\text{Qu}}_1$ may provide a window into the aromatic character of the ${\text{S}}_2$ LEAs and, therefore, a tractable path for exploration of photophysical processes.

In the PA series, the bicyclic parent ⁵ Np ^0B undergoes dramatic geometric distortion from planarity and displays one strongly diatropic ring ($-26$ ppm) and one essentially atropic ring ($-4$ ppm; current density shows no ring current). Elongation results in dramatic changes, including a return to planarity and symmetry. In ⁵ Np ^1B, the global current weakens and is replaced by relatively strong local paratropic currents in the terminal rings (28 ppm; Figure 5). ⁵ Np ^2B retains a more uniform global paratropic current, resulting in moderate paratropicity throughout (18 and 13 ppm in the terminal and central rings, respectively). In the longer ⁵ Np ^3B and ⁵ Np ^4B, localized currents are observed, with the penultimate rings sustaining robust paratropic currents (⁵ Np ^3B: 20; ⁵ Np ^4B: 15 ppm), while the terminal rings start to develop diatropic currents (⁵ Np ^3B: $-3$; ⁵ Np ^4B: $-13$ ppm), and the central subunits are essentially atropic (⁵ Np ^3B: $-5$; ⁵ Np ^4B: $-4$ ppm).

FIGURE 5.

Magnetic aromaticity and spin density plots of ${\text{Qu}}_1$‐PAs and ${\text{Qu}}_1$‐LEAs. (A) Current density plots depicted at 1 Å above the molecular plane. (B) $\text{NICS}{(1)}_{\pi zz}$values (ppm). (C) Spin density plots depicted at the isosurface value $\alpha =0.02.$ The rings with the greatest spin density are highlighted in pink. Double bonds omitted for clarity.

In the LEA series, the bicyclic parent ⁵ Az ^0B sustains a robust 10$\pi$e global diatropic current (5MR: $-18$; 7MR: $-24$ ppm). The global current is lost in ⁵ Az ^1B  which exhibits a semi‐global diatropic region (7MR: $-10$; 6MR: $-9$ ppm) alongside a local paratropic region (5MR: 7 ppm). Further elongation to ⁵ Az ^2B creates three distinct regions—an atropic 7MR (1 ppm), a more strongly diatropic benzenoid subunit ($-9$ and $-21$ ppm), and a more strongly paratropic 5MR (13 ppm). Current localization becomes progressively more pronounced along the series, with both ⁵ Az ^3B and ⁵ Az ^4B displaying localized currents within individual rings. In both molecules, robust paratropic currents are observed in the 5MRs (18 and 15 ppm) and the 7MRs (24 and 19 ppm), and weaker paratropic currents in the non‐adjacent 6MR (15 and 7 ppm). The penultimate 6MRs begin to develop diatropic character, which is negligible‐to‐weak in ⁵ Az ^3B but meaningful in ⁵ Az ^4B.

Thus, despite originating from fundamentally different magnetic responses, both families converge toward localized current patterns upon elongation, albeit with opposite spatial polarity: the termini in LEAs and the penultimate rings in PAs, similarly to the behavior observed in ${\text{T}}_1$. Once again, the most paratropic rings coincide with the regions of highest spin density, consistent with Baird's rule (Figure 5C). Unlike the ${\text{S}}_0$ and ${\text{T}}_1$ states, the ${\text{Qu}}_1$ magnetic response combines features previously observed separately: Terminal localization characteristic of the LEAs and a PA‐like benzenoid core. This observation forms the basis for the unified resonance‐structure analysis presented below.

3.3. RS‐Based Rationalization

Having characterized the magnetic aromaticity and spin density of both series across ${\text{S}}_0$, ${\text{T}}_1$, and ${\text{Qu}}_1$, we now turn to a unifying rationalization grounded in RSs and graph‐theoretical concepts—Clar's rule [69, 70], Glidewell–Lloyd's rule [90], Randić's conjugated circuit method [71, 72, 73], and Mandado's analysis of spin‐polarized systems [74]—that accounts for the behavior of both families across all three spin states.

According to Randić's model, the overall magnetic response of a molecule can be predicted by summing the conjugated circuits (closed circuits of alternating single and double bonds) within each RS of a molecule, wherein $4n+2$ $\pi$e circuits are diatropic and $4n$ $\pi$e are paratropic. The Mandado analysis extends this idea to OS cases by counting $\alpha$ and $\beta$ electrons separately and identifying circuits of $2n+1$ (diatropic) and $2n$ (paratropic) $\pi$e counts. Because paratropic currents are generally stronger than diatropic ones in similar‐sized rings, opposing currents in the same circuit are expected to produce weakly paratropic responses. Clar's rule states that the most contributing RS is the one that has the greatest number of disjoint sets of 6$\pi$e (i.e., Clar sextets). Glidewell and Lloyd generalize Clar's rule to non‐benzenoid systems, stating that the preference is to form the largest number of the smallest $4n+2$ circuits. In Figures 6, 7, 8 we present the Clar structures of all isomers in each of the three spin states.

FIGURE 6.

Clar structures for ${\text{S}}_0$‐PAs and ${\text{S}}_0$‐LEAs. Blue: $4n+2$/$2n+1$ circuits; Red: $4n$/$2n$ circuits; Orange: circuits having opposing character.

FIGURE 7.

Clar structures for ${\text{T}}_1$‐PAs and ${\text{T}}_1$‐LEAs. Blue: $4n+2$/$2n+1$ circuits; Red: $4n$/$2n$ circuits; Orange: circuits having opposing character.

FIGURE 8.

Clar structures for ${\text{Qu}}_1$‐PAs and ${\text{Qu}}_1$‐LEAs. Blue: $4n+2$/$2n+1$ circuits; Red: $4n$/$2n$ circuits; Orange: circuits having opposing character.

For each ${\text{S}}_0$‐PAs, it is possible to draw a CS Clar structure sustaining a single migrating Clar sextet, or an OS Clar structure sustaining two sextets (fixed or migrating, depending on the location of spin). The analyses described above demonstrate that the isomers ¹ Np ^0B–¹ Np ^3B are best described by the CS RS, indicating that the energetic cost associated with adopting an OS structure outweighs the potential stabilization. In contrast, ¹ Np ^4B does display diradical character, which results in two pairs of symmetric Clar structures (Figure 6A). The location of unpaired electrons predicted by these structures is in excellent agreement with the spin density plot shown above (Figure 3C), including the observation of greater spin density on the central rings, which allows for shorter migration paths for the sextets. More fundamentally, the results for ¹ Np ^4B suggest that, beyond six fused rings, migration of a Clar sextet is no longer sufficiently stabilizing. This length therefore marks a threshold past which adoption of diradical character to form an additional sextet becomes advantageous, consistent with previous theoretical analyses predicting the emergence of diradical and eventually polyradical character in longer acenes (tetraradical at $n_{\text{rings}}$ = 12; hexaradical at $n_{\text{rings}}$ = 21) [75, 91].

The ${\text{S}}_0$‐LEAs magnetic aromaticity can also be explained with RSs (Figure 6B). The magnetic behavior of ¹ Az ^0B is commonly rationalized by intramolecular charge transfer from the 7MR to the 5MR, to generate a zwitterionic RS consisting of a tropylium cation fused to a cyclopentadienyl anion [32, 82]. This distribution of charge creates two local 6$\pi$e circuits, which coexist with the 10$\pi$e global circuit, in excellent agreement with our current density plot above (Figure 3A; the current is stronger in the 5MR, in accordance with the Glidewell–Lloyd rule). For ¹ Az ^1B, the presence of a 6MR invokes Clar's rule, which may only be achieved by breaking a $\pi$‐bond. Heterolytic cleavage results in two RSs, each allowing the formation of a 6$\pi$e circuit in either the 5MR or 7MR. Despite their different magnetic responses, the same underlying electronic picture applies to the OS LEAs, ¹ Az ^2B–¹ Az ^4B. In these molecules, Clar's rule is achieved by homolytic $\pi$‐bond cleavage, resulting in an OS structure in which the unpaired electrons are sequestered on the terminal 5MR and 7MR, leading to opposing $\alpha$‐ and $\beta$‐electron currents (highlighted in orange in Figure 6B)—in excellent agreement with the magnetic aromaticity and spin density plots shown in Figure 3). As an additional consequence of this electron placement, the central benzenoid motifs are rendered similar to CS PAs, complete with their characteristic migrating Clar sextet, explaining the similarities in magnetic aromaticity described above.

In ${\text{T}}_1$, as in the OSS ground states, the Clar structures show that the unpaired electrons occupy the positions maximizing local aromatic stabilization (Figure 7). In the PAs this permits formation of two Clar sextets, whereas in the LEAs it preserves a migrating Clar sextet in the benzenoid core while maximizing spin separation. Consequently, the paratropic rings coincide with the sacrificed aromatic units, whereas the surviving Clar sextets remain diatropic. The additional Clar sextet in the ${\text{T}}_1$‐PAs is consistent with their relative thermodynamic stability (Figure 2A). The observation that this difference in energy grows with size may be attributed to the LEAs' single sextet migrating over increasingly longer benzenoid substructures, reducing its stabilizing effect [9]. This aligns with the framework proposed by Ottosson and coworkers, whereby a combination of Baird's and Hückel's rules is required to rationalize aromatic character and triplet‐state energies in such systems [84].

In ${\text{Qu}}_1$, each molecule must accommodate four unpaired electrons. Rings bearing minimal spin density may retain their diatropicity, but this arrives at the cost of forcing greater spin density onto a smaller subsection of the molecule, thereby increasing both local paratropicity and the repulsion between the unpaired electrons. ⁵ Np ^0B provides an extreme manifestation of this choice: The molecule is dramatically distorted from planarity and almost all spin density is placed on one ring, sacrificing its aromaticity entirely while allowing the other ring to behave like a ground state benzene (Figure 8A).

The longer PAs instead balance these competing requirements by distributing the spin density such that as many Clar sextets as possible are formed. The Clar structures for ⁵ Np ^1B show it can sustain a single sextet while distributing the spin on the two other rings. ⁵ Np ^2B can achieve two Clar sextets by placing the unpaired electrons on alternating rings. ⁵ Np ^3B and ⁵ Np ^4B can both form three Clar sextets; the former via a single Clar structure in which the spin density is distributed on the penultimate rings, the latter via three different Clar structures, two placing the spin on alternating rings and one placing them at the penultimate rings. In all cases, there is an excellent agreement between the graph‐theoretical RSs and the DFT‐calculated spin density plots seen above (Figure 5C).

The same compromise governs the ${\text{Qu}}_1$‐LEAs (Figure 8B). Two of the unpaired electrons preferentially populate the terminal 5MR/7MR end‐caps, sequestering the unpaired electrons from each other and from the benzenoid core. The remaining spin is placed on the central benzenoid substructure, which adopts a ${\text{T}}_1$‐PA‐like arrangement. In effect, the ${\text{Qu}}_1$‐LEAs exhibit a superposition of the ${\text{T}}_1$‐LEA endcap localization and the ${\text{T}}_1$‐PA central pattern. The RSs formed by this logic are in excellent agreement with the observed magnetic aromaticity trends and the computed spin densities shown above.

The RS‐based analysis therefore provides a consistent explanation for the magnetic response across all three electronic states. In every case, the preferred electronic structure is the one that preserves the greatest degree of local aromatic stabilization, with the distribution of unpaired electrons adapting to these aromatic constraints.

4. Conclusions

In this work, we systematically investigated the electronic structure and magnetic aromaticity of LEAs in their ${\text{S}}_0$, ${\text{T}}_1$, and ${\text{Qu}}_1$ states, and compared them directly with their isomeric PAs. Although, LEAs differ from PAs only by replacing the terminal benzene rings with fused 5‐ and 7MRs, this seemingly modest topological change fundamentally alters the evolution of their electronic structure with increasing molecular size.

LEAs develop OS character at substantially shorter lengths than PAs, accompanied by a rapid collapse of the ${\text{S}}_0$–${\text{T}}_1$ energy gap. This behavior arises because the terminal 5MR and 7MR provide energetically favorable reservoirs for unpaired electrons, allowing the central benzenoid framework to retain aromatic stabilization through a migrating Clar sextet. The same principle governs the excited states: Whereas spin density localizes in the central rings of ${\text{T}}_1$‐PAs, it localizes on the terminal odd‐membered rings of ${\text{T}}_1$‐LEAs. In ${\text{Qu}}_1$, the LEAs combine this terminal localization with the characteristic ${\text{T}}_1$‐PA arrangement in the benzenoid core, effectively representing a superposition of the two electronic structures.

Despite the markedly different magnetic aromaticity patterns observed for the two families, our analysis shows that they are governed by the same underlying driving force: Maximization of local aromatic stabilization. The unpaired electrons are distributed in such a way as to enable the most favorable aromatic framework, resulting in a partitioning of the molecule into aromatic and antiaromatic regions, each following a local application of Hückel's or Baird's rules. When there is more than one way to achieve the same stabilization (i.e., more than one Clar structure), then preference is observed for increasing the distance between the unpaired electrons. More broadly, these results suggest a hierarchy, whereby local aromatic stabilization is the dominant organizing principle and outweighs the cost of localized spin density and strongly paratropic regions.

A similar principle was recently identified in cycloheptatrienyl‐fused pentalenopentalenes, another family of non‐alternant hydrocarbons containing fused 5MR and 7MRs [92]. This suggests that the preference for local aromaticity is not unique to LEAs, but represents a more general feature of polycyclic $\pi$‐systems. Clar's rule therefore emerges as a remarkably robust tool, which remains predictive well beyond the CS ground state for which it was developed. LEAs thus serve not only as a model system for understanding non‐benzenoid $\pi$‐systems, but also shed additional light on the interplay between molecular topology and aromaticity, furthering the development of more general design principles.

Funding

This work was supported by the Israel Science Foundation (Grant No. 1745/23), Lady Davis Postdoctoral Fellowship, and Technion ‐ Israel Institute of Technology.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Section S1: Computational details.

Table S1: List of $\pi$‐orbitals for all molecules in the dataset in $S_0$.

Table S2: List of $\pi$‐orbitals for all molecules in the dataset in $T_1$.

Table S3: List of $\pi$‐orbitals for all molecules in the dataset in $Q_1$.

Section S2: Benchmarking.

Table S4: RMSD between optimized geometries and XRD crystal structure of azulene taken from reference S20.

Table S5: DLPNO‐CCSD(T) relative energies for geometries optimized at different levels of theory (kcal/mol).

Table S6: $⟨S^2⟩$ values obtained with different functionals.

Table S7: Yamaguchi $y_0$ values calculated using UHF and UKS formalisms on CAM‐B3LYP‐ and B3LYP‐optimized geometries.

Table S8: Frontier natural‐orbital occupation numbers and Yamaguchi $y_0$ values calculated with UHF and UKS formalisms on CAM‐B3LYP optimized geometries.

Section S3: Additional results.

Figure S1: CAM‐B3LYP/Def2‐TZVP calculated current density of ¹ Az ^0B (top). Main resonance structures (middle). Conjugated circuits associated with each RSs (bottom).

Figure S2: (A) Geometries, (B) current density plots, and (C) $\text{NICS}{(1)}_{\pi zz}$ values for ¹ Az ^0B calculated with CAM‐B3LYP, B3LYP, PBE, $\omega$B97X, and M11.

Figure S3: Geometries of ${\text{S}}_0$‐PAs.

Figure S4: Geometries of ${\text{S}}_0$‐LEAs.

Figure S5: Geometries of ${\text{T}}_1$‐PAs.

Figure S6: Geometries of ${\text{T}}_1$‐LEAs.

Figure S7: Geometries of ${\text{Qu}}_1$‐PAs.

Figure S8: Geometries of ${\text{Qu}}_1$‐LEAs.

Figure S9: Integrated bond currents for ${\text{S}}_0$‐PAs and ${\text{S}}_0$‐LEAs.

Figure S10: Integrated bond currents for ${\text{T}}_1$‐PAs and ${\text{T}}_1$‐LEAs.

Figure S11: Integrated bond currents for ${\text{Qu}}_1$‐PAs and ${\text{Qu}}_1$‐LEAs.

Table S9: Optimized geometry of ¹ Np ^0B. $E_{\text{SCF}}=-385.801482677$ a.u.

Table S10: Optimized geometry of ¹ Np ^1B. $E_{\text{SCF}}=-539.402074041$ a.u.

Table S11: Optimized geometry of ¹ Np ^2B. $E_{\text{SCF}}=-692.999636466$ a.u.

Table S12: Optimized geometry of ¹ Np ^3B. $E_{\text{SCF}}=-846.595715976$ a.u.

Table S13: Optimized geometry of ¹ Np ^4B. $E_{\text{SCF}}=-1000.19410640$ a.u.

Table S14: Optimized geometry of ¹ Az ^0B. $E_{\text{SCF}}=-385.740935347$ a.u.

Table S15: Optimized geometry of ¹ Az ^1B. $E_{\text{SCF}}=-539.313045583$ a.u.

Table S16: Optimized geometry of ¹ Az ^2B. $E_{\text{SCF}}=-692.912067229$ a.u.

Table S17: Optimized geometry of ¹ Az ^3B. $E_{\text{SCF}}=-846.5161302$ a.u.

Table S18: Optimized geometry of ¹ Az ^4B. $E_{\text{SCF}}=-1000.11629133$ a.u.

Table S19: Optimized geometry of ³ Np ^0B. $E_{\text{SCF}}=-385.698927085$ a.u.

Table S20: Optimized geometry of ³ Np ^1B. $E_{\text{SCF}}=-539.333252905$ a.u.

Table S21: Optimized geometry of ³ Np ^2B. $E_{\text{SCF}}=-692.954017254$ a.u.

Table S22: Optimized geometry of ³ Np ^3B. $E_{\text{SCF}}=-846.566611303$ a.u.

Table S23: Optimized geometry of ³ Np ^4B. $E_{\text{SCF}}=-1000.17389286$ a.u.

Table S24: Optimized geometry of ³ Az ^0B. $E_{\text{SCF}}=-385.681578102$ a.u.

Table S25: Optimized geometry of ³ Az ^1B. $E_{\text{SCF}}=-539.299038997$ a.u.

Table S26: Optimized geometry of ³ Az ^2B. $E_{\text{SCF}}=-692.911633803$ a.u.

Table S27: Optimized geometry of ³ Az ^3B. $E_{\text{SCF}}=-846.515947365$ a.u.

Table S28: Optimized geometry of ³ Az ^4B. $E_{\text{SCF}}=-1000.11625206$ a.u.

Table S29: Optimized geometry of ⁵ Np ^0B. $E_{\text{SCF}}=-385.563229068$ a.u.

Table S30: Optimized geometry of ⁵ Np ^1B. $E_{\text{SCF}}=-539.181903166$ a.u.

Table S31: Optimized geometry of ⁵ Np ^2B. $E_{\text{SCF}}=-692.824288923$ a.u.

Table S32: Optimized geometry of ⁵ Np ^3B. $E_{\text{SCF}}=-846.462357267$ a.u.

Table S33: Optimized geometry of ⁵ Np ^4B. $E_{\text{SCF}}=-1000.09021044$ a.u.

Table S34: Optimized geometry of ⁵ Az ^0B. $E_{\text{SCF}}=-385.550550110$ a.u.

Table S35: Optimized geometry of ⁵ Az ^1B. $E_{\text{SCF}}=-539.171541510$ a.u.

Table S36: Optimized geometry of ⁵ Az ^2B. $E_{\text{SCF}}=-692.802001650$ a.u.

Table S37: Optimized geometry of ⁵ Az ^3B. $E_{\text{SCF}}=-846.430317402$ a.u.

Table S38: Optimized geometry of ⁵ Az ^4B. $E_{\text{SCF}}=-1000.05403752$ a.u.

Additional details on choice of computational level of theory and benchmarking, input templates for all forms of calculations performed, integrated bond currents, xyz coordinates for all structures.

Data Availability Statement

The data underlying this study are available in the published article and in its Supporting Information.