Magnetism in nanographenes attracts a lot of attention due to its unique properties such as weak spin-orbit coupling and long spin coherence time,1,2 compared with conventional magnetism originating from d- or f-block elements. With the advanced developments of on-surface synthesis,3-9 nanographenes can be precisely fabricated with well-defined \(\pi\)-electron topologies upon the design of molecular precursors,10-13 which are ideal models for investigating carbon-based magnetism and the underlying mechanism of magnetic exchange interaction.14 To induce magnetism in nanographenes, embedding substitutional heteroatoms15-18 or incorporating pentagon rings19-23 have been employed in nanographenes upon on-surface synthesis. Besides these approaches, creating sublattice imbalance in bipartite lattices can also generate net spins as predicted by Ovchinnikov’s rule and Lieb’s theorem.24,25 Accordingly, triangulene and its \(\pi\)-extended homologues with sublattice imbalance have been fabricated on surface,26-28 which show high-spin ground states in agreement with theory predictions. For realizing spin-logic devices or molecular switches at room temperature,29,30 robust magnetic ordering with large magnetic coupling strength is needed. Towards this goal, the largest magnetic exchange coupling of 102 meV has been reported in rhombus-shaped nanographenes with zigzag periphery. 31

Besides the interests in the intrinsic properties of triangulenes and rhombus nanographenes, the more attractive feature is that they can serve as building blocks for constructing spin networks with collective quantum behaviors to explore novel quantum phases of matter.32–39 The linkers, connecting adjacent nanographene units, have played a key role in engineering the magnetic exchange coupling among building blocks. As reported, antiferromagnetic coupling can be tuned to ferromagnetic coupling through adjusting the carbon-carbon bonding sites from the same or opposite sublattices.12,33,40,41 The magnetic exchange interaction strength has also been tuned by including a connection spacer42 or by incorporating a pentagon ring for tailoring singly occupied orbital overlap at the connecting region.35 However, the effects of orbital symmetry on the magnetic exchange interaction have been rarely addressed. Especially, the symmetry of frontier orbitals, the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), has been found critical for both chemical and physical processes. For example, a cycloaddition reaction requires the symmetry of HOMO to be the same as the symmetry of LUMO.43 Furthermore \(\pi\)-conjugated polymer’s quantum topological phase transition is observed to show an avoided level crossing of HOMO and LUMO.44 Considering the \(\pi\)-magnetism in nanographenes, different levels of theoretical interpretations45–47 have unveiled the role of frontier orbitals in promoting the emergence of unpaired electrons. Inspired by a theoretical work which reported that molecular orbital symmetries are crucial for determining the exchange coupling of diradical nanographenes,48 we implemented the symmetry of frontier orbitals in the design of a nanographene with widely tunable magnetic exchange interactions.

In this work, we employ phenalenyl and two different linkers (Fig. 1a) as the building blocks to fabricate various nanographene dimers with different conjugation symmetry (Fig. 1b) by using an on-surface synthesis approach. Theoretically, according to Lieb’s theorem,25 dimers **D3** and **D6** have sublattice imbalance thus hosting a net spin of S = 1. For the other dimers, although they have balanced sublattices, they may also exhibit open-shell structures by comparing their Kekulé with non-Kekulé structures, whose energy cost of having radicals could be compensated by the extra Clar sextets (Fig. 1c).49 In experiments, upon scanning tunnelling spectroscopy (STS) and inelastic electron tunnelling spectroscopy (IETS) measurements, we find that dimers **D4** and **D5** indeed show local magnetic moments with antiferromagnetic coupling. Particularly, their exchange interactions can be widely tuned from 20 meV to 160 meV. As revealed by different levels of theoretical calculations, we attribute this variability to their different frontier orbital symmetries. In addition, we found that distant-neighbor hopping also significantly affect magnetic exchange interaction by including the third nearest-neighbor hopping effect in the calculations. Moreover, we further reveal the magnetic ground state and excited states of a spin-1/2 trimer with unequal magnetic coupling strengths, consistent with Heisenberg spin chain model calculations. Our results reveal a new approach for effectively tailoring the exchange interaction strength in nanographenes through engineering frontier orbital symmetry, which is inspiring for both theoretical design and practical realization of room-temperature spintronic devices.

The magnetic ground states of all the six dimers have been addressed by calculations at different levels. Since the ferromagnetically coupled **D3** and **D6** configurations are well understood by theory and experiments, we focus on the competing cases of **D1**, **D2**, **D4** and **D5** as depicted in Fig. 2. All the cases have an open-shell S = 0 ground state and an S = 1 excited state, suggesting the energy cost by introducing two radicals is smaller than the energy gain of adding two Clar sextets (cf. Figure 1c). The spin density distributions are shown in Fig. 2a, with spin-down density localizing at the left side and spin-up density at the right side. Although they share the same ground and excited magnetic states, their exchange coupling strengths vary dramatically as shown in Fig. 2b. **D4** hosts a coupling strength around 20 meV, much smaller than those of the remaining three cases. For **D1**/**D4**, density functional theory (DFT) calculations agree well with mean-field Hubbard (MFH) calculations with U = 3.5 eV considering only nearest-neighbor hopping. However, MFH coupling strengths of **D2**/**D5** are significantly smaller than those obtained by DFT. We noticed that distant-neighbor hopping may play an important role in exchange coupling strength of **D2**/**D5**. By adding a third-nearest-neighbor hopping term of *t*3 = -0.4 eV in MFH calculations, the obtained results quantitatively agree with DFT. To address the enhanced electron-electron interactions in such small nanographenes, complete active space Hubbard (CAS-Hubbard) calculations50 have been performed29. For **D1**,**D2** and **D5**, the CAS-calculated coupling strengths are much larger than our mean-field calculations (suggesting an enhanced Coulomb repulsion in these nanographenes); while for **D4**, the values are very close to or even smaller than those of our mean-field calculations.

Figure 3 depicts the physical nature behind such large-scale variations of magnetic coupling strength. The MFH-calculated wave functions of HOMO and LUMO are shown in Fig. 3a (the Coulomb term U is set to zero, and the nanographenes are in a closed-shell configuration). The blue and red isosurfaces denote the opposite signs of wave functions, while the dashed lines represent the mirror planes normal to the \(\pi\) systems. The HOMOs of **D1**/**D4** are antisymmetric with respect to the mirror plane, while the HOMOs of **D2**/**D5** are symmetric. We notice that the frontier orbital symmetry reversal changes the orbital density overlap at the linkers, and thus modifies the exchange coupling. As highlighted in Fig. 3b, the only difference between the two linkers is that linker 2 has one more phenyl ring along the mirror plane direction than linker 1. Although the difference is tiny from the chemical point of view, this influences the magnetic exchange interaction. For the **D2**/**D5** with symmetric HOMOs, since their orbital waves mainly extend perpendicular to the mirror plane direction, the addition of an extra ring along the mirror plane does not affect the orbital density distribution at the connecting region (highlighted in Fig. 3b). However, for the **D1**/**D4** with antisymmetric HOMOs, since their orbital waves can extend along the mirror plane direction, the addition of one phenyl ring in linker 2 can effectively reduce the singly-occupied orbital density per site at the connecting region. As a consequence, **D4** has a much smaller coupling strength than **D1**/**D2/D5** due to the reduced orbital density overlap at the linkers, originating from frontier orbital symmetry differences.

Orbital symmetry does not only affect orbital density distributions at the linkers, but also affects the effective coupling through the distant-neighbor hopping effect. This behavior is presented by plotting wave functions in the defined form of \({\psi }_{L,R}={\psi }_{HOMO}\pm {\psi }_{LUMO}\) as shown in Fig. 3c. This suggests that in all cases \({\psi }_{L,R}\) are spatially separated, localized at the left or right side of the dimer. The distant-neighbor hopping effect on the coupling strength can be captured by considering the effective hopping *t*3 between \({\psi }_{L}\) and \({\psi }_{R}\), in the definition of \(<{\psi }_{L}\left|{\widehat{H}}_{t3}\right|{\psi }_{R}>\) (The next-nearest hopping *t*2 of graphene honeycomb lattice is less important in our case (cf. supplementary information)). At the connecting region, the orbital sign at two sites connected by *t*3 is the same for **D1**/**D4** with antisymmetric HOMOs, but opposite for **D2**/**D5** with symmetric HOMOs (as marked by red/blue arrows in Fig. 3c). The change of HOMO-LUMO energy gap depends on \({<\psi }_{L}\left|{\widehat{H}}_{t3}\right|{\psi }_{R}>\), which is negative for **D1**/**D4** and positive for **D2**/**D5** considering the value of *t*3 is negative for graphene systems.51 As shown in Fig. 3d, the HOMO-LUMO energy gap of **D4** (**D5**) decreases (increases) with the increased magnitude of *t*3, suggesting a reduced (enhanced) effective coupling strength. As a result, the magnetic exchange coupling in phenalenyl dimers can be widely engineered by tuning frontier orbital symmetries, enabled by the third-nearest-neighbor hopping effect.

To fabricate these different types of nanographene dimers experimentally, precursor **1** in Fig. 4a is designed as an 8-methyl-naphthalene and a benzene ring with two bromide substitutes. Precursor **1** was first deposited on Au(111) with submonolayer coverage and then annealed to the temperatures of 433 K and 523 K subsequently for triggering debrominative cycloaddition and cyclodehydrogenation reactions. As the reaction scheme shows in Fig. 4a, due to the adsorption handedness, the [2 + 2 + 2] cycloaddition reaction52,53 of three precursors **1** with the same or opposite adsorption handedness will lead to different products **2** and **3** formed by three phenalenyl units conjugated by the additional formed phenyl ring in different orientations. Figure 4b shows the overview image of the main products **2** and **3** by using scanning tunneling microscopy (STM), most of which are passivated by two hydrogen atoms on the phenalenyl units during the on-surface synthesis process.12,40 Upon tip manipulation, the desired diradical dimers as well as the trimer **T** with three radicals can be obtained by a voltage pulse at different passivation sites. These *sp**3* passivated diradical dimers host the same low-lying spin states as **D4**, **D5** and **D6** in Fig. 3 as suggested by MFH calculations (Supplementary Fig. 13,14). Hereafter, we also refer to these dimers as **D4**, **D5** and **D6** for convenience. Their chemical structures are further characterized by bond-resolved atomic force microscopy (AFM) measurements,54,55 which are shown sequentially in Fig. 4c,d. In the AFM images of dimers **D4**, **D5** and **D6**, the remaining hydrogen passivation site can be observed due to the bright contrast from the additional hydrogen atom repulsion.

The theoretically predicted magnetic exchange coupling for nanographene dimers **D4** and **D5** are verified in experiments by spin-flip spectroscopy. The AFM images and their corresponding chemical structures are shown in Fig. 5a. One of the three phenalenyl units is passivated by two hydrogen atoms, thus having two unpaired electrons in the \(\pi\) system. d*I*/d*V* spectroscopy is measured on different positions of **D4-D6** as marked in the AFM images (Fig. 5b). For dimer **D4**, symmetric steps around the Fermi level are detected at \(\pm\) 20 meV, indicating an inelastic excitation.11,22 We attribute these as spin-flip features corresponding to the singlet-triplet spin excitation, giving an antiferromagnetic coupling of 20 meV, which agrees quantitatively well with the CAS-Hubbard model calculations with *t*3 = -0.4 eV and U = 4.1 eV. By sharp contrast, the corresponding d*I*/d*V* spectra taken on **D5** exhibit a substantially increased excitation threshold of 160 meV, consistent with the CAS-Hubbard model calculations. These results experimentally prove that the exchange interaction can be widely engineered by tuning the frontier orbital symmetry (Fig. 3). Dimer **D6**, as predicted by Lieb’s theorem in Fig. 1, should have an open-shell triplet ground state, which has been confirmed by a Kondo resonance showing a peak feature at the Fermi level in d*I*/d*V* spectra.12,41 To visualize the spatial distribution of the spin excitation and the Kondo effect, d*I*/d*V* maps recorded at the excitation energies of \(\pm\) 20 meV, \(\pm\) 160 meV, and the Fermi energy are shown in Fig. 5c, respectively. They all agree well with the simulated STM images using the singly-occupied orbitals as shown in Fig. 5c, further confirming that the studied nanographene dimers have an open-shell ground state.

In addition, a spin trimer can also be studied with our molecules. All extra hydrogen atoms can be dissociated by voltage pulses, thus resulting in a trimer composed of three radicals (Fig. 6a-c). Among these three radical sites, two of them have ferromagnetic interaction, while the other two pairs have different strengths of antiferromagnetic exchange interactions. The unequal exchange interactions are curious and compete in this trimer system. d*I*/d*V* spectra are measured at different radical sites as marked in Fig. 6a. As shown in Fig. 6d, a Kondo feature is detected at the Fermi level at position 3, while the spin-flip features with an excitation threshold of 160 meV are observed at positions 1 and 2. The spin-excitation d*I*/d*V* maps and the Kondo map are shown in Fig. 6e. These results suggest that this spin trimer has a ground state of S = 1/2. To understand this system, a Heisenberg spin trimer model (Fig. 6f) is solved, demonstrating that S = 1/2 is the ground state and the other six excited states are nearly degenerate with an excitation gap of 160 meV, consistent with experiments (Fig. 6g). CAS-Hubbard calculations have been performed to address spin correlations among these spins. The energy gaps between the ground state of S = 1/2 and the first excited state of S = 3/2 are illustrated as a function of U/|t| in Fig. 6h. The excitation gap between S = 1/2 and S = 3/2 is 170 meV considering U/|t| = 1.3 comparable with Heisenberg spin chain model calculations. The spin correlators among the three spins are calculated through the term \(<\varPsi \left|{S}_{z}\left(i\right){S}_{z}\left(j\right)\right|\varPsi >\), with \(\varPsi\) the ground-state wave function and \({S}_{z}\) the spin operator. As shown in Fig. 6i, the calculated spin correlator maps show the antiferromagnetic spins at the top-right and bottom are strongly correlated, while the spin at the top-left is almost isolated from the others, in agreement with experiments.

In summary, we demonstrate an approach of engineering the magnetic-exchange interaction in \(\pi\)-conjugated nanographenes upon tuning the frontier orbital symmetries. Combined with different levels of theoretical calculations and scanning probe microscope measurements, the chemical and electronic structures of various phenalenyl dimers have been investigated on surfaces. As the theoretical calculations predicted, their coupling strengths can be widely tuned from 20 meV to 160 meV, upon flipping the HOMO/LUMO symmetry. Moreover, the competition among three radicals with unequal magnetic coupling strengths has been demonstrated, in agreement with Heisenberg spin chain model calculations. Our results provide insights for engineering magnetic-exchange interaction in a large-scale combined frontier orbital symmetry with covalent bridging, which could extend the design strategy for realizing graphene-based spintronic materials in future above room temperature.