Probing Local Aromaticity in π-Stacked Polyfluorenes Using New Descriptors Based on the Electron Density

Probing Local Aromaticity in π-Stacked Polyfluorenes Using New Descriptors Based on the Electron Density | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 26 November 2025 V1 Latest version Share on Probing Local Aromaticity in π-Stacked Polyfluorenes Using New Descriptors Based on the Electron Density Authors : Matheus Máximo-Canadas , Nathália M. P. Rosa , and Itamar Borges 0000-0002-8492-1223 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176413535.51712063/v1 166 views 94 downloads Contents Abstract Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Pi-stacked systems stabilize molecular structures and enable efficient charge transport and energy dissipation in functional materials, and also provide a framework for investigating local aromaticity under spatial confinement. In this work, we apply the recently developed Q 2 -based aromaticity descriptors ( ACS Omega 2025 , 10, 14, 1415) to stacked systems. They are derived from components of the electric quadrupole second-rank tensor Q 2 of the distributed multipole analysis (DMA) expansion, the first to include out-of-plane electron contributions. We investigated pi-stacked polyfluorene systems having one to four layers ( F 1 - F 4 ), and the Q 2 -based aromaticity descriptor trends were compared with the traditional indices NICS, PDI, FLU, and HOMA . The Q 2 ± 1 and NICS ± 1 indices are affected by interlayer couplings, thereby not accurately describing aromaticity trends. On the other hand, the descriptors Q 2 0 and Q 2 ring atoms accurately reproduced aromatic and antiaromatic ring trends and showed that local aromaticity remains nearly unchanged when increasing the number of layers from F 1 to F 4, which is consistent with the PDI, FLU, and HOMA indices. Probing Local Aromaticity in π-Stacked Polyfluorenes Using New Descriptors Based on the Electron Density Matheus Máximo-Canadas a , Nathália M. P. Rosa a , Itamar Borges Jr a,* a Departamento de Química, Instituto Militar de Engenharia (IME), Praça General Tibúrcio, 80, Rio de Janeiro, RJ 22290-270, Brasil * Email: [email protected] Abstract Pi-stacked systems stabilize molecular structures and enable efficient charge transport and energy dissipation in functional materials, and also provide a framework for investigating local aromaticity under spatial confinement. In this work, we apply the recently developed \(\mathbf{Q}_{2}\) -based aromaticity descriptors ( ACS Omega 2025 , 10, 14, 1415) to stacked systems. They are derived from components of the electric quadrupole second-rank tensor \(\mathbf{Q}_{2}\) of the distributed multipole analysis (DMA) expansion, the first to include out-of-plane electron contributions. We investigated pi-stacked polyfluorene systems having one to four layers ( \(\mathbf{F}\mathbf{1-F}\mathbf{4}\) ), and the \(\mathbf{Q}_{2}\) -based aromaticity descriptor trends were compared with the traditional indices NICS , PDI , FLU , and HOMA . The \(\left|\mathbf{Q}_{2}\right|\left(\pm 1\right)\) and \(\text{NICS}\left(\pm 1\right)\) indices are affected by interlayer couplings, thereby not accurately describing aromaticity trends. On the other hand, the descriptors \(\left|\mathbf{Q}_{2}\right|\left(0\right)\) and \(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) accurately reproduced aromatic and antiaromatic ring trends and showed that local aromaticity remains nearly unchanged when increasing the number of layers from \(\mathbf{F}\mathbf{1}\) to \(\mathbf{F}\mathbf{4}\) , which is consistent with the PDI , FLU , and HOMAindices . Keywords: π-stacked polyfluorenes; Local aromaticity;\(\mathbf{Q}_{2}\)-based aromaticity descriptors; Distributed Multipole Analysis (DMA). Introduction Pi-stacked systems are molecular arrangements in which two or more π-rich units associate spatially, one above another. 1 Depending on the electronic interactions and the surrounding environment, these systems can adopt Face-to-Face, T-Shaped, and Parallel-Displaced (or offset) configurations, 2 depicted in FIGURE 1 - they are of great significance for biological systems. The structure and stability of DNA rely not only on hydrogen bonding between complementary bases but also on π-stacking between adjacent bases along the same strand, 3 i.e., the stacked DNA Watson-Crick bases. 4 FIGURE 1 Different types of π-stacked arrangements. From left to right: Face-to-Face, T-Shaped or Parallel-Displaced (or offset) systems. Pi-stacked aromatic rings maintain attractive interactions even under large lateral displacements, which supports their spontaneous organization into stable structures in proteins, crystals, and supramolecular materials. 5 This intrinsic stability arises from a balance of dispersion, steric repulsion, and quantum charge penetration, and favors practical applications. 6 Specifically, in energetic materials, π-π stacking enables layer sliding and energy dissipation in explosives, reducing localized hotspots and enhancing safety. 7 The pi-stacked systems also have remarkable versatility in technological applications: their arrangement supports the coexistence of magnetism and electrical conduction within a single structure, facilitating molecular spintronic devices. 8 Furthermore, π-π interactions enable efficient through-space electronic communication, enhancing the performance of OLEDs, 9 nanoelectronics, 10 and organic solar cells, 11 allowing more compact designs than traditional π-conjugated systems. 12 Beyond their structural and technological significance, π-stacked systems offer an exceptional model system to investigate how aromaticity behaves in spatially confined environments. Since π-π interactions arise from the overlap of delocalized π orbitals, the intrinsic aromatic character of each unit governs both the strength and the nature of the stacking. 13 Additionally, the proximity of neighboring π clouds can perturb local ring currents and modify the aromatic or antiaromatic character of the involved units. 14 Therefore, understanding how aromaticity is preserved, attenuated, or redistributed upon stacking can help to elucidate the electronic structure and the properties of these systems. Although nearly two-thirds of all known chemical compounds are aromatic species, 15 a unified definition of aromaticity still does not exist. Additionally, aromaticity cannot be defined quantum-mechanically as a directly measurable observable. 16 Instead, aromaticity is a theoretical concept created to explain experimental behaviors characteristic of a particular class of compounds, 17 which include the favored bond-length equalization, remarkable thermodynamic stability, distinctive magnetic signatures such as induced ring currents, and a peculiar chemical reactivity that favors substitution reactions over addition. In this way, aromaticity serves as a unifying idea, bringing conceptual coherence to experimental phenomena that would otherwise appear unclear. 18 Initially confined to the realm of classical organic molecules, aromaticity has since been generalized far beyond its original scope. 19 Today, this concept includes a wide variety of nontraditional systems that have led to the identification of numerous types of aromatic compounds, including Möbius, 20 homo-, 21 hetero-, 22 Clar, 23 σ-, 24,25 metallo-, 26 three-dimensional aromaticity, 27 and others. 28 Considering this diversity, many attempts have been made to establish quantitative measures that capture the degree of aromatic character via aromaticity descriptors or indices. Therefore, the classification of a molecule as aromatic or nonaromatic depends mainly on the theoretical definition adopted. 29 These descriptors rely on different physical principles, including electronic delocalization, 30 energetic stabilization, 31 bond-length equalization, 32 or magnetic response. 33 Among the descriptors based on magnetic criteria, 33–35 the Nucleus-Independent Chemical Shift (NICS) probes the presence of induced electronic ring currents indirectly by calculating the magnetic shielding at selected points in or above the molecular plane; different variants (e.g.,\(NICS(0)\), \(NICS(1)\), and the\(\text{NICS}\left(1\right)_{\text{zz}}\)) are often used to capture the sensitivity to π currents better. 36,37 From a structural perspective, the Harmonic Oscillator Model of Aromaticity (HOMA) quantifies the uniformity of bond distances within a ring. 38 In the electronic domain, the Fluctuation Index (FLU) 39 is employed to assess π-electron delocalization and multicenter correlation by measuring fluctuations in electron-sharing indices between adjacent bonds, and the para -Delocalization Index (PDI) 40 evaluates the extent of electron delocalization between atoms at para positions within a ring. Because aromaticity manifests through multiple, often interdependent phenomena, no single descriptor can fully capture its nature. Different metrics generally provide different perspectives; therefore, the choice of a descriptor depends on the specific focus of the problem. As noted previously, 41 such indices must be interpreted with caution, as a rigorous quantitative assessment of aromaticity should rely exclusively on experimentally measurable molecular response properties. Indices lacking experimental correspondence and falsifiability are therefore epistemologically inadequate for defining or quantifying aromaticity. Although it may be an oversimplification to attribute aromaticity just to the delocalization of π-electrons perpendicularly to the molecular plane, this factor remains central to its manifestation. 19,30 Motivated by this understanding, our group has recently proposed a new family of six aromaticity descriptors 42 based on the components of the quadrupole moment tensor \(\mathbf{Q}_{2}\) from Stone’s Distributed Multipole Analysis (DMA) electric multipole expansion. 43,44 In the DMA formalism, the electron density is partitioned into electric multipoles localized on specific sites throughout the molecule. The proposed descriptors rely on the components of the second-order tensor \(\mathbf{Q}_{2}\), which is the first term in the multipole expansion that includes out-of-plane electron contributions. 45 These indices have proven effective across twelve sets of aromatic and antiaromatic molecules in the benchmark introduced by the Solà group, 19,46 showing strong consistency with established aromaticity indices. This approach has successfully captured substituent effects on the aromaticity of benzene derivatives 47,48 and the aromaticity of polycyclic aromatic hydrocarbons (PAHs). 49 In this work, we apply the \(\mathbf{Q}_{2}\)-based descriptors to another set of molecular systems. Since the π-stacked architectures reported by Solà and co-workers pose significant challenges to the magnetic descriptor NICS, 50 we aim to determine whether the \(\mathbf{Q}_{2}\)-based descriptors can provide a reliable measure of aromatic trends in these complex, multi-ring environments. Methods A series of π-stacked polyfluorenes composed of one (\(\mathbf{F}\mathbf{1}\)) to four (\(\mathbf{F}\mathbf{4}\)) fluorene units arranged in a helical stacking configuration was investigated. Each fluorene moiety contains two six-membered rings (6-MRs) fused by a five-membered ring (5-MR). The rings are labeled from \(\mathbf{A}\) to\(\mathbf{F}\): rings \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\)correspond to the first layer in all molecules, while \(\mathbf{D}\),\(\mathbf{E}\), and \(\mathbf{F}\) belong to the second layer in the\(\mathbf{F}\mathbf{2}\), \(\mathbf{F}\mathbf{3}\), and\(\mathbf{F}\mathbf{4}\) systems. These systems were synthesized by Rathore et al. , 51 and their chemical structures are presented in FIGURE 2. FIGURE 2 Line representation of the investigated π-stacked systems: \(\mathbf{F}\mathbf{1}\), \(\mathbf{F}\mathbf{2}\),\(\mathbf{F}\mathbf{3}\) , and \(\mathbf{F}\mathbf{4}\), comprising one, two, three, and four layers, respectively, each containing two six-membered rings fused by a five-membered ring. The rings analyzed in this work are labeled from \(\mathbf{A}\) to\(\mathbf{F}\) to facilitate discussion of local aromaticity. To enable a rigorous comparison with other aromaticity descriptors reported in the literature, the optimized geometries of the four systems (\(\mathbf{F}\mathbf{1}\), \(\mathbf{F}\mathbf{2}\),\(\mathbf{F}\mathbf{3}\) , and \(\mathbf{F}\mathbf{4}\)) were taken from the work of the Solà group, 50 which computed the NICS, PDI, FLU, andHOMA descriptors for these systems. Professor Solà kindly supplied the geometries, which are available in the Supplementary Material for reference. They optimized the singlet ground state\(\left(S_{0}\right)\) employing the B3LYP/6-31G* method in the syn conformation without symmetry constraints. Although this method represents a relatively modest level of theory today and does not include dispersion corrections, it yielded results consistent with the X-ray structures and previous ab initio studies. For comparison purposes, the geometry of benzene was optimized using the same methodology. Single-point energy calculations were subsequently performed using Gaussian 16 Revision A.03 to obtain the electron densities required for evaluating the \(\mathbf{Q}_{2}\)-based aromaticity descriptors, using the same level of theory of the geometry optimization (B3LYP/6-31G*). Following the protocol established in our previous work, in which these descriptors were introduced, 42 two keywords were included in the Gaussian input files: nosymm and Density=Current . The nosymm keyword prevents any symmetry-based reorientation of the molecular frame during the single-point calculation. The Density=Current keyword instructs Gaussian to use the electron density from the converged DFT calculation, ensuring that computed properties are derived from the correct electron density. The Gaussian-formatted checkpoint files ( fchk ) were then used to partition the electron density using the GDMA2 software, which calculates the Distributed Multipole Analysis (DMA). This is an accurate approach for partitioning molecular charge distributions into localized electric multipole moments, representing the electron density as a multicenter expansion of monopoles, dipoles, quadrupoles, and higher-order terms, with the multipole electric moments positioned at atomic centers and predefined sites. The overall magnitude of the quadrupole moment at a given molecular position is the Euclidean norm of its five independent spherical tensor components, as shown in Equation 1: \({|\mathbf{Q}}_{2}|=\ \left(\sum_{k}\left|\mathbf{Q}_{2k}\right|^{2}\right)^{\frac{1}{2}}=\sqrt{{(Q_{20})}^{2}+{(Q_{21c})}^{2}+{(Q_{21s})}^{2}+{(Q_{22c})}^{2}+{(Q_{22s})}^{2}}\) Eq. (1) Here, \(Q_{20}\), \(Q_{21c}\), \(Q_{21s}\), \(Q_{22c}\), and \(Q_{22s}\)represent the components of the \(Q_{2}\) tensor in spherical coordinates, which are related to the projections along the \(x,\ y\), and \(z\) directions of a Cartesian coordinate system – they are defined from Equation 2 to Equation 6: \(Q_{21c}=\frac{2}{\sqrt{3}}\Theta_{\text{xz}}\) Eq. (3) \(Q_{21s}=\frac{2}{\sqrt{3}}\Theta_{\text{yz}}\) Eq. (4) \(Q_{22s}=\frac{2}{\sqrt{3}}\Theta_{\text{xy}}\) Eq. (5) \(Q_{22c}=\frac{1}{\sqrt{3}}\left(\Theta_{\text{xx}}-\Theta_{\text{yy}}\right)\) Eq. (6) where \(\Theta_{\text{ij}}\) denote the Cartesian components of the\(\mathbf{Q}_{2}\) tensor. The out-of-plane component (\(Q_{2_{\text{zz}}}\)) expectation value can be expressed in integral form as: \(\left\langle Q_{2_{\text{zz}}}\right\rangle=Q_{20}=\int{\rho\left(r\right)R_{20}\left(r\right)\text{dr}}\mathbb{=C}\int{\rho\left(r\right)r^{2}\left(\frac{3}{2}\cos^{2}\theta-\frac{1}{2}\right)\text{dr}}\) Eq. (7) where \(\rho\left(\mathbf{r}\right)\) is the electron density,\(\theta\) is the polar angle relative to the \(z\)-axis, and\(\mathbb{C}\) is a normalization constant. The set of six \(\mathbf{Q}_{2}\)-based aromaticity descriptors is defined in Table 1. The full description and computational workflow for calculating the aromaticity indices are detailed in our original publication. 42 TABLE 1 Aromaticity descriptors derived from the\(\mathbf{Q}_{2}\) DMA tensor components. Tensor component values are in atomic units (\(ea_{0}^{2}\)) and can be normalized to the corresponding value for benzene. \(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) The total sum of the \(\left|\mathbf{Q}_{2}\right|\) values of each ring atom \({Q_{2}}_{zz,\ \ ring\ atoms}\) The out-of-plane quadrupole tensor components \(Q_{20}={Q_{2}}_{\text{zz}}\) values of each ring atom \(\left|\mathbf{Q}_{2}\right|_{\text{origin}}\) The total sum of the \(\left|\mathbf{Q}_{2}\right|\ \)values computed for all the atoms of a molecule referred to \(r_{\text{origin}}=\left(0,\ 0,\ 0\right)\) \({Q_{2}}_{zz,\ \ origin}\) The out-of-plane quadrupole tensor components \(Q_{20}={Q_{2}}_{\text{zz}}\) values of all the atoms of a molecule referred to \(r_{\text{origin}}=\left(0,0,0\right)\) \(\left|\mathbf{Q}_{2}\right|\left(\alpha\right)\) The magnitude of \(\mathbf{Q}_{2}\) computed at \(r=\left(0,\ 0,\ \alpha\right)\) a \({Q_{2}(\alpha)}_{\text{zz}}\) The out-of-plane quadrupole tensor components \(Q_{20}=Q_{\text{zz}}\) computed at \(r=\left(0,\ 0,\ \alpha\right)\) a a In this work, three positions along the z -axis were considered for each 6- or 5-membered ring, namely:\(\alpha\) = 1 Å, 0 Å, and \(-\)1 Å. Consequently, for each ring (with labels from \(\mathbf{A}\) to \(\mathbf{F}\)), the corresponding values \(|\mathbf{Q}_{2}|(1)\), \(|\mathbf{Q}_{2}|(0)\),\(|\mathbf{Q}_{2}|(-1)\), and \(Q_{2}\left(1\right)_{\text{zz}}\),\(Q_{2}\left(0\right)_{\text{zz}}\),\(Q_{2}\left(-1\right)_{\text{zz}}\) were reported. Adapted from ACS Omega, 2025 (https://doi.org/10.1021/acsomega.4c11451) under the terms of the CC BY 4.0 license. For each analyzed five- and six-membered ring, three reference points were defined along the \(z\)-axis to compute the\(|\mathbf{Q}_{2}|(\alpha)\) and \({Q_{2}(\alpha)}_{\text{zz}}\)descriptors, for \(\alpha\ =\ -1\ Å\), \(0\ Å\), and \(+1\ Å\), respectively, relative to the geometric center of each ring. The stick representations of the investigated π-stacked systems are shown in FIGURE 3, where these \(\alpha\) sites are pictured. FIGURE 3 The stick representations of the π-stacked systems (from \(\mathbf{F}\mathbf{1}\) to \(\mathbf{F}\mathbf{4}\)) highlight the three reference points (\(\alpha=-1\ Å\), \(0\ Å\), and\(+1\ Å\)) defined along the \(z\)-axis relative to the geometric center of each five- and six-membered ring. The colored spheres indicate the positions used for the calculation of the\(|\mathbf{Q}_{2}|(\alpha)\) and \({Q_{2}(\alpha)}_{\text{zz}}\)descriptors: red for \(\alpha=-1\ Å\), green for \(\alpha=0\ Å\), and blue for \(\alpha=+1\ Å\). To allow comparisons among the different systems, we employed normalized values of the \(\mathbf{Q}_{2}\)-based descriptors. They are obtained by dividing each descriptor value by the corresponding value for benzene, the reference system, the prototypical aromatic system. In the following discussion, only the normalized values are considered. Results and discussion The complete set of computed aromaticity descriptors is summarized in Tables S1 to S3 of the Supplementary Material. Table S1 collects the\(\mathbf{Q}_{2}\)-based aromaticity descriptor values for benzene and the π-stacked polyfluorenes (\(\mathbf{F}\mathbf{1-F}\mathbf{4}\)). For direct comparison across systems of different sizes and stacking degrees, Table S2 collects the normalized values. Finally, Table S3 groups the normalized data by equivalent ring labels (\(\mathbf{A}-\mathbf{F}\)) to facilitate the analysis of local aromaticity within each layer. For each table, other aromaticity indice values (PDI, FLU, HOMA, andNICS) 50 are also presented. The normalized \(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\)values in Table S2 reveal a pronounced aromatic character for the six-membered rings (6-MRs). The aromaticity values remain remarkably close to that of benzene, exhibiting normalized values between \(0.90\)and \(0.97\), where benzene itself is defined as \(1.00\). In contrast, the five-membered rings (5-MRs) display a decrease in aromaticity, with\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) values around\(0.70\), indicating a less aromatic (antiaromatic) character. The\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) descriptor, which quantifies aromaticity at the geometric center of each ring, confirms this picture: it shows even higher aromatic character for the 6-MRs, with typical values near \(0.95\), while the 5-MRs reach only about\(0.80\). This behavior is further confirmed by molecular electrostatic potential (MEP) maps, where the electron density is mainly located at the centers of the 6-MRs – see the top-view panels in FIGURE 4. In contrast, the five-membered ring (5-MR) in the central part of the structures shows less pronounced electron density at its center. This behavior is entirely consistent with previous report in the literature, 50 in which NICS,PDI, FLU, and HOMA descriptors collectively describe the same aromatic trend. FIGURE 4 Molecular electrostatic potential (MEP) maps of the investigated molecules, from top to bottom:\(\mathbf{F}\mathbf{1}\), \(\mathbf{F}\mathbf{2}\),\(\mathbf{F}\mathbf{3}\), and \(\mathbf{F}\mathbf{4}\). The electrostatic potential values are given in hartrees with isovalue\(=0.001\) hartree. The red regions indicate electron-rich areas (negative MEP values), while the blue regions correspond to electron-deficient areas (positive MEP values). Three representations of each molecule are shown, from left to right: front, side, and top views. Regarding the influence of the number of layers on the aromaticity, both\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) descriptors show that the aromatic character remains essentially invariant as the system size increases from \(\mathbf{F}\mathbf{1}\) to \(\mathbf{F}\mathbf{4}\) – see values in Table S3. This observation applies to both the inner and outer rings. For\(\left|\mathbf{Q}_{2}\right|\left(0\right)\), variations are practically negligible (about 0.001), whereas\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) values show only minor fluctuations (about 0.02). Such stability across different system sizes matches the behavior observed in the PDI,FLU, and HOMA descriptors, all of which suggest that aromatic delocalization is largely preserved and unaffected by the number of layers. This behavior, however, cannot be directly inferred from the\(\left|\mathbf{Q}_{2}\right|\left(1\right)\) and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\) values – there are inconsistent values across equivalent rings (\(\mathbf{A}-\mathbf{F}\)) in Table S3. For the outer layer (rings\(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) for all analyzed molecules), the \(\left|\mathbf{Q}_{2}\right|\left(-1\right)\)descriptor remains consistent as the number of layers increases, as these points lie below the molecule and these rings are not affected by any inner-layer interactions (i.e., by the rings \(\mathbf{D}\),\(\mathbf{E}\), and \(\mathbf{F}\); see the green sites in FIGURE 3). Conversely, the \(\left|\mathbf{Q}_{2}\right|\left(1\right)\) values for the outer layer of systems \(\mathbf{F}\mathbf{2}\),\(\mathbf{F}\mathbf{3}\), and \(\mathbf{F}\mathbf{4}\) (rings\(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\)), points, which are located at the interface between the outer and the subsequent inner layer (the blue sites in FIGURE 3), display non-uniform variations as the number of layers increases. A similar inconsistency trend is observed for the inner layer (rings\(\mathbf{D}\), \(\mathbf{E}\), and \(\mathbf{F}\)). For the\(\mathbf{F}\mathbf{1}\) and \(\mathbf{F}\mathbf{2}\) systems, the\(\left|\mathbf{Q}_{2}\right|\left(1\right)\) values remain stable due to the absence of upper layers. However, upon inclusion of further layers in the \(\mathbf{F}\mathbf{3}\) and \(\mathbf{F}\mathbf{4}\)systems, \(\left|\mathbf{Q}_{2}\right|\left(1\right)\) and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\) begin to fluctuate irregularly, reflecting the increasing electronic coupling between adjacent layers. This non-linear evolution clearly indicates that both\(\left|\mathbf{Q}_{2}\right|\left(1\right)\) and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\) descriptors are sensitive to interlayer interactions and the progressive stacking of aromatic units. Visualization of the MEPs (FIGURE 4) supports this interpretation. As the number of layers increases, a clear overlap of the electrostatic potentials between successive layers becomes evident in the front and side representations. This interlayer overlap explains the observed deviations in \(\left|\mathbf{Q}_{2}\right|\left(1\right)\) and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\) values and confirms that the aromaticity character mapped by these two descriptors is subtly modulated by the proximity and electronic influence of neighboring π-systems. Therefore, the \(\mathbf{Q}_{2}\)-based descriptors exhibit problematic coupling effects when additional points are placed in very close proximity. Multipole moments interact through the electrostatic field, such that a given site may experience short-range coupling to another site. When charge distributions begin to overlap, the multipole expansion introduces an error known as charge penetration or penetration energy. 45 In the present π-stacked systems, no explicit overlap of additional sites occurs, i.e., there is no overlap between the green and the blue sites in FIGURE 3. Nevertheless, spatial proximity alone can perturb the multipolar expansion, leading to inconsistencies in the \(\mathbf{Q}_{2}\) values for out-of-plane points as the number of stacked layers increases. This behavior is analogous to the known limitations of theNICS descriptor, where \(NICS(1)\) and \(NICS(-1)\) values often become unreliable under similar conditions – see the inconsistent values across equivalent rings (\(\mathbf{A}-\mathbf{F}\)) in Table S3. For the present systems, the NICSanalysis 50 indicates an apparent increase in aromaticity as the system size expands from \(\mathbf{F}\mathbf{1}\) to\(\mathbf{F}\mathbf{4}\) for both inner and outer rings. Notably, a more substantial increase is detected for ring \(\mathbf{C}\) compared to ring \(\mathbf{A}\) for \(\mathbf{F}\mathbf{2}\) to\(\mathbf{F}\mathbf{4}\), a trend previously attributed to enhanced magnetic coupling effects in the vicinity of ring\(\mathbf{C}\). 50 The same inconsistency, i.e., a disproportionate rise in aromaticity at the ring \(\mathbf{C}\), was also detected by \(\left|\mathbf{Q}_{2}\right|\left(1\right)\), thus reinforcing the notion that both electrostatic and magnetic coupling mechanisms contribute to these anomalies. In the optimized geometries of the π-stacked systems, the \(\mathbf{C}\)rings lie closer to the atoms of the adjacent layers than the\(\mathbf{A}\) rings, leading to stronger interlayer magnetic and electrostatic couplings. Since the magnetic response in NICSis distance-dependent, the coupling is more pronounced at ring\(\mathbf{C}\), artificially enhancing its aromatic character. Therefore, unsurprisingly, both the \(\mathbf{Q}_{2}\)-based andNICS descriptors agree in their limitations: when interlayer distances approach the regime where short-range interactions or field couplings become significant, their respective assumptions, i.e., electrostatic independence for \(\mathbf{Q}_{2}\) and magnetic one forNICS, no longer hold, leading to deviations in the aromaticity character. This effect becomes even more pronounced as the stacking increases to three and four layers, i.e., for molecules \(\mathbf{F}\mathbf{3}\) and\(\mathbf{F}\mathbf{4}\). In these configurations, the rings located at the center of the stack are positioned between two adjacent layers. This geometric arrangement intensifies magnetic and electrostatic couplings, since the central layer is simultaneously influenced by the magnetic fields or electrostatic potentials generated above and below it. As a consequence, the NICS values become increasingly negative, apparently suggesting a higher degree of aromaticity. However, this enhancement is artificial rather than real, arising from the superposition of induced magnetic fields rather than from the genuine π-electron delocalization typical of aromaticity. 50 For the \(\left|\mathbf{Q}_{2}\right|\left(1\right)\) and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\) descriptors, the interlayer coupling produces the opposite trend, indicating a reduction in aromaticity. For these reasons, \(NICS(1)\), \(NICS(-1)\),\(\left|\mathbf{Q}_{2}\right|\left(1\right)\), and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\) cannot be considered reliable aromaticity descriptors in multilayered systems. Nevertheless, the \(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) descriptors successfully reproduced the same aromatic trends predicted by thePDI, FLU, and HOMA descriptors, remaining stable and consistent even in the presence of extensive stacking. This agreement reinforces the robustness of\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) as reliable indicators of in-plane aromaticity, even when inter-layer interactions become significant. For the out-of-plane components of the quadrupole tensor\(Q_{2_{zz,ring\ atoms}}\), and\(Q_{2}\left(\alpha\right)_{\text{zz}}\), although these descriptors exhibit even greater sensitivity than their analogues for identifying aromatic and antiaromatic rings, they remain inherently susceptible to geometrical distortions. In particular, their reliability depends on the alignment between the normal vector of the analyzed ring and the Cartesian z-axis. Accurate interpretation of the zzcomponents requires the ring to lie as close as possible to thexy-plane, ensuring that its normal vector is parallel to the\(z\)-axis. In the π-stacked polyfluorene series, however, the progressive increase in stacking leads to systematic deviations from coplanarity. While the first layer remains precisely aligned with thexy-plane, each additional layer becomes progressively more misaligned. As a consequence, the normal vectors of the upper layers no longer coincide with the Cartesian \(z\) axis, and the computedzz components do not represent the actual out-of-plane quadrupole contribution of each ring. Instead, they correspond only to its projection onto the z-axis. This geometric misalignment introduces artificial variations, ultimately leading to inconsistent zz values across different rings and stacking layers. Importantly, this issue does not affect \(Q_{2_{zz,origin}}\), as this descriptor is evaluated at a fixed reference point and therefore remains invariant with increasing stacking. The quadrupole moments, referenced to the global origin of the system (\(\left|\mathbf{Q}_{2}\right|_{\text{origin}}\)), normalized by the corresponding benzene descriptors, exhibit approximately linear increases with the successive addition of π-stacked fluorene units. This behavior is expected and has a predominantly geometric source. In the GDMA2 software, multipole moments are computed for distributed sites and subsequently summed relative to a fixed Cartesian origin. Consequently, the inclusion of additional layers contributes additional quadrupolar anisotropies that coherently accumulate, leading to systematic variations in the overall quadrupole values. The nearly linear growth observed for \(\left|\mathbf{Q}_{2}\right|_{\text{origin}}\) and\({Q_{2}}_{\text{zz},\ \ \text{origin}}\) descriptors thus reflect the extensive summation of electronic anisotropies rather than an effective enhancement of local aromaticity. To isolate this extensive additive effect, the total quadrupole values were divided by the number of layers, in addition to normalization by the respective benzene descriptor value – see Table S2. A systematic decrease in the average contribution per layer was then observed. This finding indicates that the linear accumulation of total descriptor values arises primarily from the addition of layers rather than from any intrinsic increase in the aromatic character of individual rings. FIGURE 5a and FIGURE 5b plot the normalized results obtained for\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and\(\left|\mathbf{Q}_{2}\right|\left(0\right)\), respectively. All the presented descriptors exhibit a consistent trend in the increase and decrease of aromaticity values across the investigated systems. A noticeable reduction in aromaticity is observed for the five-membered rings (\(B_{5}^{1}\) and \(E_{5}^{2}\)) compared to the six-membered rings, highlighting their antiaromaticity character. In FIGURE 5a, just slight variations in\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) values can be distinguished for equivalent rings (\(\mathbf{A}-\mathbf{F}\)). In contrast, FIGURE 5b reveals almost no variation in\(\left|\mathbf{Q}_{2}\right|\left(0\right)\). This behavior confirms that both\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) descriptors provide consistent and reliable measures of local aromaticity for the π-stacked polyfluorenes. Nevertheless,\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) proved to be even more robust, as it remains invariant for the equivalent rings (\(\mathbf{A}-\mathbf{F}\)). Figure S1 in the Supplementary Material shows analogous plots for\(\left|\mathbf{Q}_{2}\right|\left(1\right)\) and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\). These components, however, display irregular behavior, particularly the\(\left|\mathbf{Q}_{2}\right|\left(1\right)\) descriptor. FIGURE 5 Comparison between normalized\(\mathbf{Q}_{2}\)-based aromaticity descriptors and other aromaticity indices (PDI, FLU, HOMA, andNICS 50 ) for the π-stacked polyfluorenes (\(\mathbf{F}\mathbf{1}-\mathbf{F}\mathbf{4}\)). (a)\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and (b)\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) values are shown for each corresponding ring (\(A-F\)) for different stacking configurations. TABLE 2 summarizes the performance of the six \(\mathbf{Q}_{2}\)-based aromaticity descriptors in capturing local π-stacked aromaticity of the studied π-stacked polyfluorenes. The values of the traditionalPDI, FLU, HOMA, and NICSindices are also included. TABLE 2 Performance of the six\(\mathbf{Q}_{2}\)-based aromaticity descriptors for describing local aromaticity in the series of π-stacked polyfluorenes. A green checkmark ( ✓ ) indicates that the descriptor followed the expected aromaticity trend, and a red cross ( ✗ ) indicates a significant discrepancy. A yellow warning symbol (⚠) indicates caution regarding the choice of the reference \(\alpha\) value. \(\left|\mathbf{Q}_{2}\right|_{\text{origin}}\) ✓ Linear increase with number of layers due to additive effects; must also be normalized per number of layers. \(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) ✓ Reliable indicators of local aromaticity; reproduce trends for 6-MRs and 5-MRs; invariant across stacked layers. \(\left|\mathbf{Q}_{2}\right|\left(\alpha\right)\) ⚠ ✓ \(\alpha\ =\ 0\ Å:\) reliable; achieved the best results. ✗ \(\alpha=\pm 1\ Å:\) affected by interlayer electrostatic couplings and charge-penetration effects; unsuitable for identifying local aromaticity \({Q_{2}}_{zz,\ \ origin}\) ✓ Reliable. Not affected by coplanarity loss, as it is computed at a fixed origin rather than relative to ring-plane orientation. \({Q_{2}}_{zz,\ \ ring\ atoms}\) ✗ Cannot quantify local aromaticity in π-stacked polyfluorenes when layers are not coplanar, and when the descriptor reports projections relative to the \(z\)-axis. \({Q_{2}(\alpha)}_{\text{zz}}\) PDI ✓ Reproduce trends for 6-MRs and 5-MRs; invariant across stacked layers. FLU HOMA NICS ✗ Misleading trends due to interlayer magnetic coupling; unreliable for multilayered π-stacks. Conclusion In this work, we employed the recently introduced\(\mathbf{Q}_{2}\)-based aromaticity descriptors for a series of π-stacked polyfluorenes (\(\mathbf{F}\mathbf{1}-\mathbf{F}\mathbf{4}\)) and compared their performance against other aromaticity indices (PDI,FLU, HOMA, andNICS). 50 Our analysis shows that the\(\mathbf{Q}_{2}\)-derived indices provide a consistent probe of aromaticity effects when the appropriate descriptor is chosen; that is, using \(\left|\mathbf{Q}_{2}\right|_{\text{origin}}\) , \(\left|\mathbf{Q}_{2}\right|_{\text{ring}\ \text{atoms}}\) , and \(\left|\mathbf{Q}_{2}\right|\left(0\right)\) . B ut also reveal specific limitations when applied to the\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) and zz out-of-plane descriptors (\(Q_{2_{zz,ring\ atoms}}\),\(Q_{2_{zz,\ origin}}\), and\(Q_{2}\left(\alpha\right)_{\text{zz}}\)). Among the new descriptors,\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) proved to be robust indicators of local aromatic character for pi-stacked systems. They reproduced the expected higher aromaticity of six-membered rings, similar to benzene, and the relative reduction in aromaticity of five-membered rings, reflecting their antiaromatic character. Both descriptors remain essentially invariant as the number of stacked layers increases, consistent with trends observed for the PDI,FLU, and HOMA descriptors. In contrast, the out-of-plane descriptors\(\left|\mathbf{Q}_{2}\right|\left(1\right)\) and\(\left|\mathbf{Q}_{2}\right|\left(-1\right)\) exhibit irregular variations due to interlayer electrostatic couplings and charge-penetration effects, which compromise, in this case, their usefulness. These effects are similar to the known limitations of theNICS descriptor, which also yields misleading aromaticity trends under strong interlayer magnetic coupling. Therefore, for stacked systems \(NICS(\pm 1)\) and\(\left|\mathbf{Q}_{2}\right|\left(\pm 1\right)\) fail to provide consistent measures of aromaticity, at least for not perfectly stacked systems. The quadrupole moment, referring to the global origin of the system (\(\left|\mathbf{Q}_{2}\right|_{\text{origin}}\)) exhibits an approximately linear increase with the number of stacked layers, reflecting additive geometric effects rather than intrinsic aromatic enhancement. Normalization per number of layers and relative to benzene is therefore necessary to exclude this extensive additive effect. Although the out-of-plane quadrupole components (\(\mathbf{Q}_{2_{\text{zz}}}\)) are sensitive aromaticity probes, their reliability is limited by geometric distortions. As stacking increases and the layers (and thus the ring planes) lose coplanarity with thexy-plane, their normal vectors deviate from the Cartesian\(z\)-axis, leading to zz components representing only projected values and inconsistent results across layers. This limitation does not apply to \({Q_{2}}_{\text{zz},\ \ \text{origin}}\), since the descriptor is computed at a fixed origin rather than relative to ring-plane orientation. Overall, our findings establish that\(\left|\mathbf{Q}_{2}\right|_{\text{ring\ atoms}}\) and\(\left|\mathbf{Q}_{2}\right|\left(0\right)\) descriptors are the most reliable new descriptors for assessing local aromaticity in π-stacked polyfluorenes, alongside electronic- (PDI andFLU) and geometric- (HOMA) based indices. Data Availability The data that support the findings of this study are available in the supplementary material of this article. Additional datasets – including all geometry optimization and frequency files, single-point energy calculations, GDMA input and output files, molecular electrostatic potential (MEP) maps, and the complete spreadsheet (.xlsx) containing all the aromaticity descriptors are publicly available on GitHub and Zenodo and can be accessed at https://doi.org/10.5281/zenodo.17673633. Author Contributions Matheus Máximo-Canadas – investigation, data curation, formal analysis, visualization, first draft, writing – review and editing. Nathália M. P. Rosa – investigation, data curation, formal analysis, visualization, writing – review, and editing. Itamar Borges Jr – conceptualization, methodology, data curation, formal analysis, visualization, validation, funding acquisition, project administration, resources, supervision, writing – review, and editing. Conflicts of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments Matheus Máximo-Canadas thanks CAPES for a PhD scholarship. Nathália M. P. Rosa thanks FAPERJ (grant number E-26/205.922/2022) for a post-doctoral scholarship. I. B. thanks the Brazilian agencies CNPq (Grant number 300281/2025-0) and FAPERJ (Grant number E-26/204.294/2024) for funding this research. The authors thank Professor Miquel Solà for providing the optimized geometries used in this work. References 1. C. R. Martinez and B. L. Iverson, Chem Sci , 2012 , 3, 2191. 2. K. Molčanov and B. Kojić-Prodić, IUCrJ , 2019 , 6, 156–166. 3. M. Swart, T. van der Wijst, C. Fonseca Guerra and F. M. Bickelhaupt, J Mol Model , 2007 , 13, 1245–1257. 4. J. D. Watson and F. H. C. Crick, Nature , 1953 , 171, 737–738. 5. D. B. Ninković, J. P. Blagojević Filipović, M. B. Hall, E. N. Brothers and S. D. Zarić, ACS Cent Sci , 2020 , 6, 420–425. 6. K. Carter-Fenk and J. M. Herbert, Physical Chemistry Chemical Physics , 2020 , 22, 24870–24886. 7. R. Bu, Y. Xiong and C. Zhang, Cryst Growth Des , 2020 , 20, 2824–2841. 8. Y. Tsuji, K. Okazawa, K. Kurino and K. Yoshizawa, The Journal of Physical Chemistry C , 2022 , 126, 3244–3256. 9. K. Madhusudana Rao, R. Ramaraghavulu, D. Kolli, S. H. Babu, S. V. P. Vattikuti and M. Godumala, J Mater Chem C Mater , 2025 , 13, 3091–3122. 10. Y. Zhang, Y. Zhang, L. Wang, C. Zhang and H. Chen, Trends Chem , 2025 , 7, 299–316. 11. J. Zhang, F. Bai, Y. Li, H. Hu, B. Liu, X. Zou, H. Yu, J. Huang, D. Pan, H. Ade and H. Yan, J Mater Chem A Mater , 2019 , 7, 8136–8143. 12. S. Yang, Y. Qu, L. Liao, Z. Jiang and S. Lee, Advanced Materials , 2022 , 34. 13. E. M. Cabaleiro-Lago and J. Rodríguez-Otero, ACS Omega , 2018 , 3, 9348–9359. 14. K. Okazawa, Y. Tsuji and K. Yoshizawa, J Phys Chem A , 2023 , 127, 4780–4786. 15. G. Merino, M. Solà, I. Fernández, C. Foroutan-Nejad, P. Lazzeretti, G. Frenking, H. L. Anderson, D. Sundholm, F. P. Cossío, M. A. Petrukhina, J. Wu, J. I. Wu and A. Restrepo, Chem Sci , 2023 , 14, 5569–5576. 16. M. Solà, F Chem , 2017 , 5, 1. 17. R. Hoffmann, Am Sci , 2015 , 103, 18. 18. H. Szatylowicz, P. A. Wieczorkiewicz and T. M. Krygowski, Sci , 2022 , 4, 24. 19. F. Feixas, E. Matito, J. Poater and M. Solà, Chem Soc Rev , 2015 , 44, 6434–6451. 20. Z. S. Yoon, A. Osuka and D. Kim, Nat Chem , 2009 , 1, 113–122. 21. R. V. Williams, Chemical Reviews , 2001 , 101, 1185–1204. 22. M. Solà, A. I. Boldyrev, M. K. Cyrański, T. M. Krygowski and G. Merino, in Aromaticity and Antiaromaticity: Concepts and Applications , John Wiley & Sons, Ltd, New Jersey, 2022 , pp. 165–192. 23. A. T. Balaban and D. J. Klein, J Phys Chem C , 2009 , 113, 19123–19133. 24. A. N. Alexandrova and A. I. Boldyrev, J Phys Chem A , 2003 , 107, 554–560. 25. S. Furukawa, M. Fujita, Y. Kanatomi, M. Minoura, M. Hatanaka, K. Morokuma, K. Ishimura and M. Saito, Commun Chem , 2018 , 1, 60. 26. F. Feixas, E. Matito, J. Poater and M. Solà, Wiley Interdiscip. Rev.: Comput Mol Sci , 2013 , 3, 105–122. 27. J. Poater, M. Solà, C. Viñas and F. Teixidor, Angew Chem , 2014 , 126, 12387–12391. 28. M. Solà, in Encyclopedia of Physical Organic Chemistry , ed. Z. Wang, Weinheim: John Wiley & Sons, 2017 , vol. 6, pp. 511–542. 29. M. Solà and D. W. Szczepanik, Pure and Applied Chemistry , 2025 , 97, 1149–1157. 30. M. Solà, A. I. Boldyrev, M. K. Cyrański, T. M. Krygowski and G. Merino, in Aromaticity and Antiaromaticity: Concepts and Applications , John Wiley & Sons, Ltd, New Jersey, 2022 , pp. 145–163. 31. M. Solà, A. I. Boldyrev, M. K. Cyrański, T. M. Krygowski and G. Merino, in Aromaticity and Antiaromaticity: Concepts and Applications , John Wiley & Sons, Ltd, New Jersey, 2022 , pp. 111–130. 32. M. Solà, A. I. Boldyrev, M. K. Cyrański, T. M. Krygowski and G. Merino, in Aromaticity and Antiaromaticity: Concepts and Applications , John Wiley & Sons, Ltd, New Jersey, 2022 , pp. 77–109. 33. M. Solà, A. I. Boldyrev, M. K. Cyrański, T. M. Krygowski and G. Merino, in Aromaticity and Antiaromaticity: Concepts and Applications , John Wiley & Sons, Ltd, New Jersey, 2022 , pp. 131–143. 34. S. G. Patra and N. Mandal, International Journal of Quantum Chemistry , 2020 , 120. 35. J. E. Barquera‐Lozada, International Journal of Quantum Chemistry , 2019 , 119. 36. Z. Chen, C. S. Wannere, C. Corminboeuf, R. Puchta and P. von R. Schleyer, Chemical Reviews , 2005 , 105, 3842–3888. 37. A. Stanger, European Journal of Organic Chemistry , 2020 , 2020, 3120–3127. 38. J. Kruszewski and T. M. Krygowski, Tetrahedron Letters , 1972 , 13, 3839–3842. 39. E. Matito, M. Duran and M. Solà, The Journal of Chemical Physics , 2005 , 122, 014109. 40. J. Poater, X. Fradera, M. Duran and M. Solà, Chemistry – A European Journal , 2003 , 9, 400–406. 41. P. Lazzeretti, Physical Chemistry Chemical Physics , 2004 , 6, 217–223. 42. M. Máximo-Canadas, R. S. S. Oliveira, M. A. S. Oliveira and I. Borges, ACS Omega , 2025 , 10, 14157–14175. 43. A. J. Stone, Journal of Chemical Theory and Computation , 2005 , 1, 1128–1132. 44. A. J. Stone, Chemical Physics Letters , 1981 , 83, 233–239. 45. A. J. Stone, The Theory of Intermolecular Forces , Oxford University Press, Oxford, 2nd edn., 2013 . 46. F. Feixas, E. Matito, J. Poater and M. Solà, Journal of Computational Chemistry , 2008 , 29, 1543–1554. 47. N. M. P. Rosa, M. Máximo-Canadas and I. Borges Jr, Journal of Physical Organic Chemistry , 2025 , 38, e70024. 48. M. Máximo‐Canadas, N. M. P. Rosa and I. Borges, Journal of Computational Chemistry , 2025 , 46. 49. G. A. Salles, P. R. C. Magalhães, J. R. Carvalho, M. Máximo-Canadas, N. M. P. Rosa, J. C. V. Chagas, L. F. A. Ferrão, A. J. A. Aquino, I. Borges, F. B. C. Machado and H. Lischka, Chemistry (Easton) , 2025 , 7, 178. 50. S. Osuna, J. Poater, J. M. Bofill, P. Alemany and M. Solà, Chemical Physics Letters , 2006 , 428, 191–195. 51. R. Rathore, S. H. Abdelwahed and I. A. Guzei, Journal of the American Chemical Society , 2003 , 125, 8712–8713. Information & Authors Information Version history V1 Version 1 26 November 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords distributed multipole analysis (dma) local aromaticity π-stacked polyfluorenes Authors Affiliations Matheus Máximo-Canadas Instituto Militar de Engenharia View all articles by this author Nathália M. P. Rosa Instituto Militar de Engenharia View all articles by this author Itamar Borges 0000-0002-8492-1223 [email protected] Instituto Militar de Engenharia View all articles by this author Metrics & Citations Metrics Article Usage 166 views 94 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Matheus Máximo-Canadas, Nathália M. P. Rosa, Itamar Borges. Probing Local Aromaticity in π-Stacked Polyfluorenes Using New Descriptors Based on the Electron Density. Authorea . 26 November 2025. DOI: https://doi.org/10.22541/au.176413535.51712063/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. Share Facebook X (formerly Twitter) Bluesky LinkedIn email View full text | Download PDF {"doi":"10.22541/au.176413535.51712063/v1","type":"Article"} Now Reading: Share Figures Tables Close figure viewer Back to article Figure title goes here Change zoom level Go to figure location within the article Download figure Toggle share panel Toggle share panel Share Toggle information panel Toggle information panel Go to previous graphic Go to next graphic Go to previous table Go to next table All figures All tables View all material View all material xrefBack.goTo xrefBack.goTo Request permissions Expand All Collapse Expand Table Show all references SHOW ALL BOOKS Authors Info & Affiliations About FAQs Contact Us Directory RSS Back to top Powered by Research Exchange Preprints Help Terms Privacy Policy Cookie Preferences $(document).ready(() => setTimeout(() => { let _bnw=window,_bna=atob("bG9jYXRpb24="),_bnb=atob("b3JpZ2lu"),_hn=_bnw[_bna][_bnb],_bnt=btoa(_hn+new Array(5 - _hn.length % 4).join(" ")); $.get("/resource/lodash?t="+_bnt); },4000)); (function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'9febc39c18f64193',t:'MTc3OTI4NDU0MA=='};var a=document.createElement('script');a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();