Electronic spin relaxation in triangulene-based nanographenes

Vol.:(0123456789) Theoretical Chemistry Accounts (2025) 144:70 https://doi.org/10.1007/s00214-025-03223-3 RESEARCH Electronic spin relaxation intriangulene‑based nanographenes AliciaOmist1,2· RobertoA.Boto2 Received: 14 March 2025 / Accepted: 23 June 2025 © The Author(s) 2025 Abstract Graphene nanoribbons are versatile platforms for developing novel magnetic phases with applications in emerging technologies. Compared to molecular magnets based on transition-metal complexes, weak spin-orbit couplings (SOC) and hyperfine interactions (HFI) in carbon-based materials offer the perspective of developing magnetic states with prolonged spin coherences. In this work, we study the spin dynamics in solution for a series of open-shell triangulene-based nanographenes ([3]-rhombene and Clar’s goblet) in their lowest-energy triplet state. Developing the Redfield equations for triplet states enables the identification of the contribution of the SOC, HFI and zero-field splitting decoherences to the electronic spin relaxation. We perform density functional theory simulations to evaluate the impact of the length of the nanoribbons on the magnetic properties and thus on the relaxation times. We demonstrate that the magnetic properties of the selected nanoribbons are determined by the strong localization of the unpaired electrons on the triangulene units. Additionally, we show that for weak magnetic fields the spin dynamics are governed by HFI decoherences, but for stronger fields the SOC decoherences also play a role. The strong contribution of HFI to spin dynamics indicates the possibility of chemical substitution influencing the electronic spin dynamics. Keywords Nanoribbons· Triangulene· Magnetic properties· Redfield theory· Electronic spin relaxation time· Density functional theory 1 Introduction The manipulation of electronic spins has immense implications in spintronics and quantum technologies[1]. The application of electronic spins to data manipulation requires precise control of the spin-decoherence channels, which are mainly determined by spin-orbit and hyperfine couplings. These couplings are weak in carbon-based materials, which offer improved platforms to study spin-polarized transport, and electronic spin qubit generation and manipulation[2]. The last decade has witnessed an impressive evolution in the synthesis of graphene fragments at the nanoscale (nanographenes and nanoribbons), due to the progress of on-surface synthesis[3]. Central to these synthetic methods is the activation of unconventional reaction mechanisms that enable access to structures with a wide variety of sizes and shapes that affect their electronic and magnetic properties[2, 4–10]. Open-shell graphene nanofragments exhibit 𝜋 -magnetism (magnetic states that fundamentally emerge from unpaired electrons in p orbitals), with magnetic states spatially localized at the edges of the nanofragment. Thus, magnetic interactions between the electrons localized at the edges fully determine the magnetism of these materials. For example, nanographenes terminated by zigzag edges exhibit 𝜋 -magnetism, but armchair edgesare not magnetic[8, 11–13]. Access to the electronic spin dynamics of graphene nanoribbons has been achieved, showing that they exhibit relaxation times of some microseconds[13–15]. Spin dynamics in these materials is determined by spin-phonon coupling (interaction between spin and vibrational degrees of freedom) that modulates spin-orbit and hyperfine interactions[16–20]. In solution and non-crystalline * Roberto A. Boto [email protected]g Alicia Omist [email protected] 1 Polimero eta Material Aurreratuak: Fisika, Kimika eta Teknologia Saila, Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), PK 1072, 20080Donostia-SanSebastián, Euskadi, Spain 2 Donostia International Physics Center (DIPC), 20018Donostia-SanSebastián, Euskadi, Spain Theoretical Chemistry Accounts (2025) 144:70 70 Page 2 of 10 solids, other relaxation mechanisms are possible involving molecular rotations and translations[21–25]. Here, we address the magnetic properties and relaxation times for some selected canonical graphene nanoribbons (Fig.1) in solution. These structures consist of triangulene dimers derived from the rhombus-shape nanographene[26, 27], and the olympicene dimer Clar’s goblet[28, 29]. We use density functional theory (DFT) to compute the magnetic properties for the lowest-energy triplet state of the selected nanoribbons. Additionally, we apply Redfield theory[30] to evaluate the contribution of the spin-orbit coupling, hyperfine interaction and zerofield splitting to the spin-relaxation process. This paper is organized as follows. In Section2, we apply Redfield theory to obtain the expressions for the relaxation times for a triplet state. In Section3, we describe the computational methodology employed. In Section4, we first discuss the evolution of the magnetic properties with the size of the nanoribbons. Specifically, we focus on the Zeeman splitting, hyperfine interaction and zero-field splitting. We then discuss the impact of these magnetic properties on the relaxation times. In the last section, we present the main conclusion of this work. 2 Theory andmethodology 2.1 Spin relaxation We address here the electronic spin relaxation process that determines the lifetime of electronic spin states. The time evolution of the expectation value of the spin operator   S in the basis of states {�𝛼⟩} is given by, where   Sp is the p-th element of the electronic spin operator, and 𝜌𝛼𝛼 ′ ≡⟨𝛼�𝜌 �𝛼′⟩ stands for the electronic spin density. Equation1 shows that the time evolution of ⟨  S p⟩ is connected with the rate equation of the spin density matrix, which strongly depends on the coupling between magnetic states and other molecular degrees of freedom. Here, we consider that the electronic spin relaxation results from the interaction between the magnetic states and the molecular rotations and diffusion. This situation is often observed for molecules in solution[21, 24]. To describe this process, we consider the Hamiltonian, (1) d dt ⟨  Sp ⟩ = � 𝛼,𝛼 � � d 𝜌 𝛼𝛼� dt �⟨ 𝛼� �  Sp � 𝛼 ⟩, (2)  H=  H 0+  H 1(t), Fig. 1 Spin distribution for the selected open-shell graphene nanofragments. a, b Schematic representation of the studied Clar’s goblet nanoribbons (CGn; panel a), and [3]-rhombene nanoribbons ([3]- Rhon; panel b). c, d 0.02 Bohr −3 isosurface of the spin density of the triplet ground state for CG1 c, and [3]-Rho1 d, where the regions colored blue and red represent regions of accumulation of spin 𝛼 and spin 𝛽 electrons, respectively. The spin density (panels c-d) is computed within DFT at the UM06-2X/6-31 G(d,p) level. The spheres colored white and gray in panels c–d represent the atoms of hydrogen and carbon, respectively Theoretical Chemistry Accounts (2025) 144:70 Page 3 of 10 70 where  H0 is a time-independent term that dominates the energy splitting of the magnetic states, and the term  H1(t) is a time-dependent contribution that accounts for the coupling between the magnetic states and the molecular motions that induce the relaxation process. Redfield theory [30] connects the rate equation of the density matrix in Eq.1 with the term  H1(t) , where R𝛼𝛼 ′ ,𝛽𝛽′ is the relaxation matrix that accounts for the decay of the element 𝛼,𝛼′ of the density matrix. R𝛼𝛼 ′ , 𝛽𝛽′ is often written in terms of the spectral function J𝛼𝛼 � ,𝛽𝛽 � (𝜔) , which informs on the correlation of  H1(t) evaluated at time t and t�=t+𝜏 , with 𝜔𝛼 , 𝛽 the angular frequency corresponding to the transition between the magnetic states 𝛼 and 𝛽 . The evaluation of Eq.4 requires averaging over ensembles of molecules, as indicated by the overbar symbol. In terms of J𝛼𝛼 � ,𝛽𝛽 � (𝜔) , the relaxation matrix reads[30]. To proceed further, we need to specify the dependence of  H1(t) on the electronic spin operator   S . Thermal motions of magnetic molecules in solution modulate the magnetic interactions of the spin density with the environment. These interactions produce fluctuations in the energy transition between the magnetic states, which are determined by Zeeman splittings (ZE), hyperfine interactions (HFI), and for spin systems with S > 1 ∕2 , by zero-field splittings (ZFS). These fluctuations are accounted for by the time-dependent Hamiltonian In the following, we consider that  H1 (t ) fluctuates randomly in the presence of a magnetic field oriented parallel to the z-axis, and thus molecules experience isotropic perturbations that propagate in different directions (see Ref. 25 for further discussion). The first term on the right-hand side of Eq.6 accounts for the fluctuation in the magnetic energy levels in the presence of an external magnetic field  B . (3) d𝜌 𝛼𝛼� dt =i �[𝜌, H0]𝛼𝛼�+∑ 𝛽,𝛽 � R𝛼𝛼�,𝛽𝛽�𝜌𝛽𝛽� , (4) J 𝛼𝛼�,𝛽𝛽�(𝜔)= ∫+∞ −∞ ⟨ 𝛼 � H1(t) � 𝛼� ⟩⟨ 𝛽� � H1(t+𝜏) � 𝛽 ⟩ e−i𝜔td𝜏 , (5) R 𝛼𝛼�,𝛽𝛽�= 1 2ℏ2 [ J𝛼𝛽𝛼�𝛽�(𝜔𝛼�𝛽�)+J𝛼𝛽𝛼�𝛽�(𝜔𝛼𝛽) −𝛿𝛼�𝛽� ∑ 𝛾 J𝛾𝛽𝛾𝛼(𝜔𝛾𝛽)−𝛿𝛼𝛽 ∑ 𝛾 J𝛾𝛼�𝛾𝛽�(𝜔𝛾𝛽�) ]. (6) H1 (t)=H (1) ZE (t)+H HFI (t)+H ZFS (t) . (7)  H( 1 ) ZE =𝜇 B  BΔgiso   S , where 𝜇B is the Bohr magneton,   S is the vectorial electronic spin operator, and Δgiso = (Δgxx +Δgyy +Δgzz)∕3 , with Δgij the ij-th element of the g-shift tensor that accounts for spin-orbit coupling (SOC) effects. In the presence of nuclei with nonzero nuclear spins, molecular thermal motions might induce time-dependent fluctuations in the interaction between electronic and nuclear spins, where A(i) iso =(A(i) xx +A(i) yy +A(i) zz )∕ 3 , and A (i) pq the pq-th element of the hyperfine coupling tensor for the nucleus i.  I (i ) z stands for the z-element of nuclear spin operator forthe nucleus i. In Eq.8, we select a nuclear configuration with all nuclear spins oriented along the z-axis, parallel to  B . Molecular motions also affect electronic spin–spin interactions for molecules with more than one unpaired electron ( S≥1 ). This situation is accounted for by the term where D is the zero-field splitting tensor, which determines the energy splitting of the magnetic sublevels in the absence of an external magnetic field. In the case of organic molecules, this energy splitting is mainly determined by direct dipolar spin–spin interactions[31]. The three interaction terms on the right-hand side of Eq.6 lead to different sources of electronic spin decoherences that drive the rate equation for the spin operator (Eq.1). Solving Eq.1 for the element of ⟨   S⟩ parallel to the external magnetic field ( ⟨  S z⟩ ) and inserting the decoherence terms (Eqs7-9) into Eq5, we obtain the longitudinal relaxation time T1 for the SOC, HFI and ZFS decoherences, where 𝜔0 is the Larmor frequency ( ℏ𝜔 0=𝜇BB0 ( ge+Δg ) , with ge the value of g for the free electron), and 𝜏c is the correlation time that determines the rate of the molecular motions that induce spin relaxation (usually a few picoseconds). D and E are the ZFS parameters D = 3 2 D zz and E = 1 2 (D xx −D yy) , Dpq the pq-th element of the tensor D . The (8)  H HFI = N � i=1   SA(i) iso ⟨  I(i) z ⟩. (9)  HZFS =   SD   S , (10) 1 T SOC 1 =2𝜇 2 B ℏ2B2 0Δg2 iso 𝜏c 1+𝜔2 0 𝜏2 c , (11) 1 T HFI 1 =2 �2 �� i A(i) iso ⟨  I(i) z ⟩�2𝜏 c 1+𝜔2 0 𝜏2 c , (12) 1 T ZFS 1 =2 5ℏ2 ( D 2 3+E2 )[ 𝜏 c 1+𝜔2 0 𝜏2 c + 4𝜏 c 1+(2𝜔0)2𝜏2 c], Theoretical Chemistry Accounts (2025) 144:70 70 Page 4 of 10 decay of ⟨   S⟩ perpendicular to the external magnetic field ( ⟨  S x,y⟩ ) is determined by the transverse relaxation time T2 , . Equations13-15 show that T2 is always shorter than T1 , but in the limit of very rapid molecular motions ( 𝜏c<< 1∕𝜔0 ), T2 converges to T1 (see discussion in Section S2 of theSupplementaryInformation(SI)). The relaxation times decrease with the strength of the perturbation for the three decoherences, but there are some differences in the elemental transitions involved in the relaxation. The relaxation for the three decoherences emerges from the combination of self-interaction processes, and electronic transitions between magnetic states that differ in one quanta of energy ( Δm=±1 ), but the ZFS decoherence also includes contributions from transitions between magnetic states with Δm=±2 . 3 Computational methods We optimize all the molecular structures within the density functional theory (DFT) framework using the program Gaussian 16 Rev. B.01[32]. We use the exchange–correlation functional UM06-2X[33] and the basis set 6-31G(d,p) to optimize the structure for the lowest-energy triplet state for some selected triangulene-based nanographenes derived from the [3]-rhombene and Clar’s goblet structures (see Fig.1). Harmonic frequency analysis does not reveal any imaginary frequency confirming that all the structures correspond to minima of the potential energy surface. We compute the magnetic properties for the optimized structures at the DFT theory level and the program ORCA 5.0[34]. The g-tensor is computed using coupled-perturbed Kohn–Sham response equations with the exchange–correlation functional B3LYP[35] and the basis set def2-SVP[36]. To compute the Zeeman term we use an external magnetic field perpendicular to the molecular plane, and parallel to the z-axis, i.e.,  B =B 0 z , with B0 the amplitude of the magnetic field. The hyperfine interaction tensor A is computed with the B3LYP exchange–correlation functional and the EPR-II basis set[37, 38]. We compute the tensor A for all the atoms of carbon and hydrogen present in the selected (13) 1 T SOC 2 =𝜇 2 B ℏ2B2 0Δg2 iso ( 𝜏c+𝜏c 1+𝜔2 0 𝜏2 c ), (14) 1 T HFI 2 =1 �2 �� i A(i) iso ⟨  I(i) z ⟩�2� 𝜏c+ 𝜏 c 1+𝜔2 0 𝜏2 c �, (15) 1 T ZFS 2 =1 5ℏ2 ( D 2 3+E2 )[ 3𝜏c+ 5𝜏 c 1+𝜔2 0 𝜏2 c + 2𝜏 c 1+(2𝜔0)2𝜏2 c]. molecules. We then scale the value of A by the corresponding isotope abundance: 1.07 % and 99.99 % for the isotopes of 13 C and 1 H, respectively. To compute the ZFS tensor D we use the PBE exchange–correlation functional[39, 40] and the basis set cc-pVDZ[41]. We compute the spin-orbit contribution to D within the framework developed by Pederson and Khanna[42]. We evaluate in Section S4 of the SI the contribution of the spin-orbit coupling to D using the Pederson–Khanna’s and the coupled-perturbed Kohn–Sham (CP-KS) formalisms[43, 44]. We show that for the selected nanoribbons, the CP-KS formalism overestimates spin-orbit coupling effects. We analyze the impact of the DFT functional and basis set on the magnetic properties in Section S3 of the SI. We show that for the selected structures, the choice of the DFT method has a negligible impact on the conclusions. To compute the relaxation time we select a value for the correlation time of 𝜏c =1 ps and for a magnetic field of amplitude 330 mT. 4 Results anddiscussion We address in the following sections the magnetic properties and spin relaxation for some selected graphene nanoribbons derived from [3]-rhombene ([3]-Rhon) and Clar’s goblet (CGn) in their lowest-energy triplet state (see Fig.1a,b). The triangulene monomers in CGn and [3]-Rhon are fused through n naphthalene and anthracene units, respectively (see Fig.1a,b and Section S1 in the SI). The first structure of the [3]-Rhon nanoribbons corresponds to the canonical [3]-rhombene structure, which does not include any linker between the triangulene monomers. The different orientations of the triangulene monomers in [3]-Rhon and CGn introduce some structural differences. For instance, the [3]- Rhon nanoribbons are planar and quite symmetric (they belong to the symmetry point group D2h ), but the structures of CG[n] are no longer planar and their symmetry is considerably reduced. This reduction in symmetry results from the steric repulsion between the monomeric units and the naphthalene linkers[45]. 4.1 Spin distribution The magnetic properties of the studied nanofragments are determined by the spatial distribution of their spin density, which strongly depends on the size and shape of the nanofragment. The ground state for CGn and [3]-Rhon is an open-shell singlet with two unpaired electrons located at the edges of the molecule, as confirmed by calculations and experimentally demonstrated[29, 46, 47]. Excitation energies computed using state-of-the-art wavefunction methods show that the first triplet state for CG0 and [3]-rhombene Theoretical Chemistry Accounts (2025) 144:70 Page 5 of 10 70 lies at ≈ 0.02 eV[46] and 1 eV[47] above the singlet ground state, respectively. Energy gaps below 1 eV are compatible with excitation energies computed for optically addressable organic molecules[48]. This suggests that the [3]-Rhon and CGn nanoribbons are promising candidates for designing organic platforms capable of hosting optically addressable magnetic states. The first triplet state exhibits similar spin distribution for all the studied nanoribbons, with the unpaired electrons mainly localized to the triangulene monomers (Fig.1c, d). For [3]-Rhon, there is a small contribution of the anthracene linkers to the spin distribution. 4.2 Magnetic properties In this section, we discuss the magnetic properties of the selected nanoribbons. We show in Eqs.10-15 that the g-shift tensor, the HFI and the ZFS tensor determine relaxation times. All these properties play different roles in the relaxation process of triplet states, inducing transitions between microstates with quantum number ms =−1, 0 and +1 . 4.2.1 Zeeman splittings Upon the action of an external magnetic field, the degeneracy of the magnetic states is lifted due to the Zeeman splitting. The energy gap between the magnetic states is proportional to the strength of the magnetic field and to the magnitude of the g-tensor, Deviations from ge are accounted for by the term 𝚫𝐠 , and mainly arise due to relativistic effects induced by (16) H ZE =𝜇B B ( ge+𝚫𝐠 )  S . theinteraction with the environment. For carbon-based materials, SOC is the main contribution to the g-shift tensor. In Fig.2b we depict the evolution of 𝚫𝐠iso with the size of the nanoribbon (d; see definition in Fig.2a). We observe that increasing the size of the nanoribbon produces a decrease of 𝚫𝐠iso , and thus the Zeeman splitting. This situation results from weakening the SOC between the unpaired electrons, which are localized to the edges of the nanoribbons (see Fig.1c,d). SOC evolves with the interelectronic distance as ≈ r −3 ij , and thus increasing d reduces SOC-related effects. The shape of the monomer (triangulene and olympicene in [3]-Rhon and CGn, respectively) plays a minor role in SOC, as reflected by the similar values of 𝚫𝐠iso for [3]-Rhon (solid red line) and CGn (dashed green line). 4.2.2 Zero‑field splittings ZFS is mainly determined by dipolar spin–spin interactions for organic molecules (second-order SOC contributions are typically negligible). We show in Fig.3 the evolution of the ZFS parameter D with the size of the nanoribbons. D rapidly decreased with the size of the nanoribbon, as the interaction between the magnetic moments, which are strongly located to the monomeric units, became weaker. The contribution of the covalent linkers to ZFS is small, but it is enough to produce some differences for [3]-Rhon and CGn. The higher ZFS values observed for [3]-Rhon may be attributed to the small participation of the linkers in the spin distribution. Despite these differences, the ZFS is small for all the selected nanoribbons, ranging from a few hundred to less than 1 MHz (see Table1). This, in conjunction with the relatively Fig. 2 Δg tensors. a Definition of the distance d (double-headed arrow) for [3]-rhombene and Clar’s goblet (CG0). b Evolution of the reduced trace of the g-shift tensor ( Δgiso ) for [3]-Rhon (solid red line, circular markers) and CGn (dashed green lines, circular markers) with d Theoretical Chemistry Accounts (2025) 144:70 70 Page 6 of 10 high values of the ratios between the parameter E and D (see Table1), reveals that the ZFS is quite isotropic. Thus, the [3]-Rhon and CGn nanoribbons lack a well-defined easy axis of magnetization. 4.2.3 Hyperfine interaction We also address the hyperfine interaction between the electronic spin density and the nuclear spins of 1 H and 13 C. Figure4 shows the evolutions of the total hyperfine interaction with the distance d for Rhon and CGn. The nuclei of 1 H and 13 C both have spin 1/2, but the contribution of 13 C to the hyperfine interaction is negligible due to its low abundance (dashed blue line). The strength of the interaction between the electronic spin density and the nuclear spins of 1 H for Rhon(Fig.4a) and CGn (Fig.4b) is quite similar and robust for all the nanoribbons’ lengths. This situation indicates that the interaction between the electronic spin density and the hydrogen atoms of the linker does not play a role on the HFI, which is mostly determined by the nuclear spin of the hydrogen atoms of the triangulene units. 4.3 Relaxation times Here, we focus on the relaxation times, which are determined by the g-shift tensor, ZFS and HFI. We show in Fig. 3 Zero-field splitting. Total ZFS parameter ( Dtotal , solid red lines, circular markers) and spin–spin contribution to Dtotal ( DSS , dashed blue lines, circular markers) as a function of the distance d for [3]-Rhon (a) and CGn (b). The values of Dtotal and DSS are depicted in absolute values. Notice the different magnitudes of the ZFS parameters, as indicated in the y-axis Table 1 Characterization of the ZFS. Eigenvalue of the ZFS tensor along the easy axis ( Deasy (in cm −1 ); second column) E/D ratio (third column) for the [3]-Rhon and CGn nanoribbons Nanoribbon Deasy E/D [3]-Rho − 0.013 0.104 [3]-Rho0 − 0.006 0.108 [3]-Rho1 − 0.002 0.105 [3]-Rho2 − 0.001 0.094 [3]-Rho3 −0.005x10 −1 0.073 CG0 − 0.002 0.095 CG1 −0.008x 10 − 1 0.035 CG2 −0.003x 10 − 1 0.004 CG3 −0.001x 10 − 1 0.023 CG4 −0.007x 10 − 2 0.057 Fig. 4 Hyperfine interactions. Total hyperfine interaction (solid red line) for [3]-Rhon (a) and CGn (b). Contribution of the nuclei of 1 H (dashed green line) and 13 C (dashed blue line) to the total HFI. The values in the y-axis are depicted in absolute values. Notice the different magnitudes of strength of the HFI, as indicated in the y-axis Theoretical Chemistry Accounts (2025) 144:70 Page 7 of 10 70 Fig.5 the relaxation times for [3]-Rhon (Fig.5a-c) and for CGn (Fig.5d-f). The relaxation process is mostly driven by the HFI decoherence, which exhibits relaxation times one and four orders of magnitude smaller than those obtained for the SOC and ZFS decoherences, respectively. THFI 1 and THFI 2 barely depend on the length of the nanoribbon. This situation is consistent with the stability of the HFI (Fig.4). Contrarily,the relaxation times for the SOC and ZFS are determined by the evolution of the SOC and the direct dipolar spin–spin interactions, which decrease with distance d. T1 (green line) and T2 (blue line) display similar values for the selected magnetic field(330 mT), but noticeable differences are observed for stronger fields (see Section S2 in the SI for further discussion). The three decoherences respond differently to the strength of the applied magnetic field, which determines the Larmor frequency 𝜔0 . The rate of the ZFS and HFI decoherences decreases with 𝜔0 , and thus relaxation times increase (see Eqs.11,12,14,15). In contrast, the relaxation times for the SOC decoherence decrease as the applied magnetic field becomes stronger ( TSOC 1,2 ∝B −1 0 ; see Eqs.10 and13), enhancing the contribution of the SOC decoherence to the relaxation process. 5 Summary andconclusions This work addresses the electronic spin relaxation process for open-shell graphene nanoribbons. We focus on the relaxation times for the lowest-energy triplet state of triangulene-based nanographenes based on [3]-rhombene ([3]-Rhon) and Clar’s goblet (CGn) in solution, where fast thermal molecular motions drive spin dynamics. To compute the relaxation times, we use Redfield theory, which connects relaxation times with fluctuations of the molecular magnetic properties, namely g-shift tensor, hyperfine interaction, and zero-field splitting. We compute Fig. 5 Relaxation times. Relaxation times for [3]-Rhon (a–c) and CGn (d–f) obtained using a magnetic field of amplitude 330 mT and a correlation time of 1ps. Contribution of the SOC decoherence (a, d), ZFS decoherence (b, e), and HFI decoherence (c, f) to the relaxation time T1 (dashed green line) and T2 (solid blue line). TX 1 and TX 2 stand for the contribution of the SOC ( X=SOC ), ZFS ( X=ZFS ) and HFI ( X=HFI ) decoherence to the relaxation times T1 and T2 , respectively. Notice the different magnitudes of the relaxation times, as indicated in the y-axis Theoretical Chemistry Accounts (2025) 144:70 70 Page 8 of 10 these magnetic properties using density functional theory (DFT). We first evaluate the evolution of the magnetic properties for [3]-Rhon and CGn with the size of the nanoribbons. The electronic spin density is strongly localized to the edges of the nanoribbons, with negligible participation of the covalent linkers. This spin distribution determines the behavior of the magnetic properties that determine the relaxation. Increasing the size of the nanoribbons drives the electronic spin system to a situation of weak coupling. Thus, Zeeman and ZFS splittings, which result from SOC and direct dipolar spin–spin interactions, respectively, close as the two monomeric units are separated. The hyperfine interaction between the electronic spin density and the nuclear spins is little affected by the length of the nanoribbons, due to the strong localization of the spin density. We then compute the contribution to the relaxation times T1 and T2 for the SOC, ZFS and HFI decoherence channels. We find that the relaxation process is driven by the HFI decoherence for standard magnetic fields used in organic spintronics. Strong magnetic fields enhance the contribution of the SOC decoherence. Further analysis demonstrates that the monomer separation barely affects the HFI contribution to the relaxation but strongly affects the contribution of the SOC and ZFS decoherence, particularly for long monomer separations. Our results suggest that harnessing the electronic spinnuclear interaction is crucial to the development of graphene-based materials. These materials aim to exploit their magnetic states to encode information for use in quantum technologies. 6 Supplementary information Minimum energy structures; Effect of the magnetic field on the relaxation times; Benchmark study of DFT functionals and basis sets for magnetic properties; Analysis of the spinorbit coupling contribution to the ZFS tensor. (PDF) Supplementary Information The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s0021402503223-3. Acknowledgements The authors thank the Spanish Ministry of Science and Innovation (MCIN/FEDER, projects PID2022-136231NB3I00), as well as the Ikur strategy under the collaboration agreement between Ikerbasque Foundation and DIPC on behalf of the Department of Education of the Basque Government. R.A.B. acknowledges the financial support from the grant awarded to the QSEIRA project (2024QUAN-000011-01) from the Gipuzkoa Quantum program’s 2024 call of the Department of Economic Promotion and Strategic Projects of the Provincial Council of Gipuzkoa. The authors are thankful for the technical and human support provided by the Donostia International Physics Center (DIPC) Computer Center. Author contributions A.O. and R. A. B. provided DFT simulations. R.A.B developed and implemented the equations for computing the relaxation times. All authors contributed to analyzing the results and writing the manuscript. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Data Availability Data sets generated during the current study are available from the corresponding author on reasonable request. Declarations Conflict of interest The authors declare no competing interests. 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