Unraveling the competition between charge and energy transfer in 0D/2D nanographene-graphene heterojunctions

Unraveling the competition between charge and energy transfer in 0D/2D nanographene-graphene heterojunctions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Unraveling the competition between charge and energy transfer in 0D/2D nanographene-graphene heterojunctions Mateusz Wlazło, Michal Langer, Oleksandr Y. Semchuk, Silvio Osella This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5199549/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 27 Dec, 2024 Read the published version in Theoretical Chemistry Accounts → Version 1 posted 9 You are reading this latest preprint version Abstract The charge and energy transfer processes in photoexcited 0D/2D donor/graphene heterojunctions occur through multiple different pathways. A donor deexcitation event occurring in the most prevalent Förster energy transfer mechanism (strongly favored over Dexter transfer in van der Waals heterojunctions) forbids the charge transfer from occurring, thus creating a competition between the two processes. By applying a robust computational approach, we describe the two processes from first principles, and quantify their rates using Förster and Marcus theories. We consider nanojunctions where the donor are nanographenes with varying size and symmetry, and discern important trends, e.g. the symmetry-induced quenching, or the enhancement due to increased size. We find that heterojunctions where nanographenes do not have a center of symmetry show decreased photoinduced hole and energy transfer rates, which can then be recovered by increasing the delocalization length, whereas for centrosymmetric nanographenes both hole and energy transfer processes are enhanced. However, the hole transfer rate dominates over the energy transfer process, providing a new computation-driven design principle for obtaining a high-charge transfer junction with minimized contribution of the competing energy transfer. Charge transfer Energy transfer Heterojunction Nanographene Computational approach Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Introduction Graphene molecules, known as nanographenes (NGs), can be synthesized with increasingly controlled shapes and edges thanks to the use of the bottom-up approach [ 1 – 4 ]. Due to the possibility of exact shape and size control, intriguing size-dependent optoelectronic properties arise when interfaced with graphene (Gr) to form 0D/2D heterojunctions [ 5 – 9 ]. To understand their function in devices such as quantum emitters [ 4 ], quantum dot transistors [ 5 ] and photodetectors [ 7 ], it is crucial to increase the knowledge of interface charge and energy transfer processes that occur at such heterojunctions. Herein, we extend the current knowledge of size-dependent charge transport properties [ 8 , 10 ] by providing additional insight from ab initio calculations, and introduce a completely new insight on the influence of NG size and symmetry on Förster resonant energy transfer (FRET), a photophysical process that is often considered to be in competition with charge transfer (CT) at low length scales of a few Angstroms [ 11 ], as is the case in vertical NG/Gr heterostructures. The nanographene/graphene (NG/Gr) interfaces have already been investigated experimentally due to their interesting light interaction properties [ 5 – 9 ]. In a study case by Wang et al. [ 8 ], the photoinduced hole transfer (PHT) was identified as the dominant CT mechanism, reconciling two competing views on the dependence of CT rate on the NG size. They suggested that van der Waals (vdW) interactions play a dominant role in determining the interfacial CT efficiency, whereas interfacial energetics and reorganization energy have a minor influence. Owing to the size-dependent increase in the interfacial coupling strength, the efficiency increased by one order of magnitude despite the decrease in the driving force of the hole transfer process. However, the mechanism behind this behavior is still elusive, prompting the current investigation via a detailed and systematic computational approach. The standard computational approach for estimation of charge or energy transfer prevalence at a heterojunction, would be based on the energy alignment of frontier orbitals. Depending on this molecular orbitals’ level alignment, two different types of junctions can be obtained, namely Type-I where the energy transfer process occur, and Type-II heterojunctions, where the charge transfer mechanism prevails. Nonetheless, in the current case where 0D heterojunctions are physisorbed on a 2D graphene monolayer surface, is not clear to foresee a priori if the prevailing process is charge or energy transfer, as the quantum confinement effect, different charge carrier localization, exciton dynamics, etc., leads to complex interfacial interactions that are not easily captured by the traditional Type-I or Type-II classifications (Fig. 1 ) [ 12 – 14 ]. To further develop the understanding of size-dependent vdW interactions in NG/Gr systems, we computationally explore the CT and FRET mechanisms (see Fig. 1 a) applied to the same set of NGs. The results shed light on whether FRET efficiency follows a similar trend as CT and contribute to the understanding of the competition between these two mechanisms. FRET is a non-radiative process of transferring energy from an excited donor species to an acceptor [ 15 ]. When two fluorescent molecules are considered, the interface can be modelled as a pair of interacting dipoles. This leads to a R -6 dependence, characteristic for all dipole-dipole interactions which take a major part in non-covalent bonding. Differences arise when low dimensional materials are considered. In fact, now a molecular dipole (a 0D material) interacting with a surface that can be approximated as an infinite plane, such as a molecule adsorbed on a Gr sheet (i.e., a material with 2D charge delocalization), differs quantitively from the simple dipolar approximation. The difference is due to the change of dimensionality of the acceptor states. The R -6 dependence is observed when the excited states of both the donor and acceptor are localized, and thus the dipolar approximation is valid. Deviations occur when the donor, or both the donor and the acceptor, have delocalized charge densities (see Fig. 1 b). In these cases, the R -6 power law softens depending on the carrier confinement dimensionality. Numerous cases of different delocalized interfaces have been derived theoretically [ 16 – 20 ] and observed experimentally [ 21 – 23 ], confirming the predicted deviations from the R − 6 law (0D/0D). For instance, a nanoparticle donor-nanowire acceptor (0D/1D) system was calculated to follow a R − 5 power law [ 19 ], a molecule on a bulk metal slab (0D/3D) a R − 3 dependence on distance [ 16 , 17 ] (verified experimentally for pyrazine on Ni(111) [ 21 ]), and two fluorescent polymer films (PFO and F8BT) transfer the energy between each other with a R − 2 dependence [ 23 ]. The derivations and experiments that are most relevant to the current case of NGs on Gr and result in a R − 4 dependence include excited molecular donors coupled to surface states in a metal [ 16 , 17 ] (verified e.g. by experiment with biacetyl molecule on silver [ 22 ]), a quantum well coupled to a quasicontinuum of high-energy quantum dot states [ 18 ] (verified in a InGaN QW/CdSe QD device), and, most recently, a fluorescent molecule on Gr [ 20 ]. Swathi et al. [ 20 ] were able to derive the localized donor/graphene energy transfer rate formula analytically within the FRET effect [ 15 ]. A few computational applications of this derivation have revealed that in molecular donor/graphene heterojunctions, the competing Dexter energy transfer mechanism is negligible compared to FRET [ 24 , 25 ], which is the focus of current work. In the current study, we combine the analytical approaches for FRET [ 15 , 26 ] and Marcus [ 27 ] for CT with ab initio computations. We aim to assess the strength of both energy and charge transfer mechanisms at a 0D/2D interfaces consisting of planar NGs with different sizes and symmetries physisorbed on a Gr monolayer (Fig. 2 ). A research question that motivates this work is to see the effect of delocalization length and change in symmetry on both CT and FRET properties, and the effect of the donor size on the interface, as it can behave less like localized donor/Gr (0D/2D) and increasingly like two delocalized planes (2D/2D), reminiscent of the interaction of two fluorescent polymer films that was shown to exhibit a R − 2 distance dependence rather than R − 4 [ 23 ]. Deriving a size limit in which delocalized interaction dominates over the localized limit will support the experimental efforts related to tunable-by-size NG optoelectronic properties. Computational details A 12×7 rectangular Gr supercell (30×30 Å in the xy plane) consisting of 336 carbon atoms was generated, and each one of the six NGs considered in this study were inserted above the graphene plane to create the heterojunction. Full geometry optimizations of the atomic coordinates of the six model heterojunctions (Fig. 2 ) were performed with first-principles calculations using planewave DFT implemented in VASP [ 28 – 31 ]. In a first optimization step, both ionic positions and cell shape was allowed to relax, and in the final one, only the zcoordinates of ionic positions of the NG were allowed to be optimized until forces acting on atoms were lower than the 0.02 eV/Å threshold. A vacuum layer of 15 Å was considered along the z-direction to avoid spurious interactions between the images of the (nano)graphene layers due to the periodic boundary conditions. The electronic exchange and correlation effects were treated by the Perdew − Burke − Ernzerhof (PBE) functional [ 32 ] with PAW-type pseudopotentials [ 33 ]. PBE-D3(BJ) theory level was applied to take into account the vdW interactions [ 34 , 35 ]. The planewave basis set cutoff of 500 eV was used, and Brillouin zone integration scheme on a 2×2×1 k-point grid. For the charge density difference and Bader charge analysis [ 36 , 37 ], calculations were carried out on a denser 4×4×1 k-point grid. Bader charge analysis was performed to evaluate the charge located on each atom (q Bader ) \(\:{q}_{Bader}={V}_{Val}-{N}_{Bader}\) (1), where V val and N Bader denote the number of valence electrons in a free atom and the computed number of valence electrons in the atom in the system, respectively. The interaction/dispersion energies E int , E disp acting between NG and Gr were evaluated as \(\:{E}_{int/disp}={E}_{AB}\left(AB\right)-{E}_{AB}\left(A\right)-{E}_{AB}\left(B\right),\) (2), where E(AB) is the total energy of the interface and E(A) and E(B) the energy of the isolated fragments with the same geometry as in the interface. Charge and energy transfer properties were derived from ground and excited state calculations implemented in Gaussian16 [ 38 ], at the DFT HSE06/6-31G(d,p) [ 39 ] functional and basis set level of theory. In order to quantify the CT rate at the interfaces, we adopt here the well-established semiclassical Marcus rate equation. In the weak coupling classical limits, the rate constant ( \(\:{k}_{if}\) ) for CT between two states (referred to as initial and final states, \(\:|i⟩\) and \(\:|f⟩\) ) can be written as: \(\:{k}_{if}=\frac{2\pi\:}{ħ}{J}_{if}^{2}\frac{1}{\sqrt{4\pi\:\lambda\:{k}_{B}T}}\text{e}\text{x}\text{p}(-\frac{\lambda\:}{4\lambda\:{k}_{B}T})\) , (3), 𝑤here \(\:{J}_{if}\) is the electronic coupling between the electronic wave functions which is computed as described using the fragment orbital method [ 40 , 41 ]. In this approach, the system is divided into fragments, in which an electron or hole is localized on a fragment and can hop from one fragment to another. The orbitals of a pair of molecules (a dimer) are projected onto a basis set defined by the orbitals of each individual molecule (the fragments). The obtained set of orthogonal molecular orbitals energies of the dimer are then used to rewrite the Fock matrix in the new localized basis set, to obtain a block-diagonal matrix. The main advantage of this method is the possibility to analyze pairs of different molecules, since it is not necessary to assume that the energies of the HOMO of the two fragments are equal, allowing for coupling from any occupied orbitals to any unoccupied ones [ 42 – 44 ]. When degenerated states are present in the system, the procedure is modified to consider the effective coupling ( \(\:{J}_{eff}\) ) computed as the square root of the sum of the squared couplings related to the number of degenerate states found in the calculation. For example, for two degenerate initial states i1, i2 and two degenerate final states f1, f2, the effective coupling is $$\:{J}_{eff}=\sqrt{{J}_{i1f1}^{2}+{J}_{i1f2}^{2}+{J}_{i2f1}^{2}+{J}_{i2f2}^{2}}$$ 4 . The reorganization energy λ is the energy cost to bring the nuclear configuration of the initial state into that of the final state while keeping the electronic configuration fixed at the initial state and was computed via the so-called four-point scheme (adiabatic potential energy surface method) [ 45 – 47 ]: $$\:{\lambda\:}_{\text{P}\text{H}\text{T}\:}=\:{E}^{+/-}\left({Q}_{N}\right)-{E}^{+/-}\left({Q}^{+/-}\right)+{E}_{N}\left({Q}^{+/-}\right)-{E}_{N}\left({Q}_{N}\right)$$ 5 , where \(\:{E}_{N}\left({Q}_{N}\right)\) and \(\:{E}^{+/-}\left({Q}^{+/-}\right)\) are the total energy of the optimized neutral and charged structures, while \(\:{E}^{+/-}\left({Q}_{N}\right)\) and \(\:{E}_{N}\left({Q}^{+/-}\right)\) are the total energy after the vertical ionizations. The coupling and reorganization energy calculations were performed at the HSE06/6-31G(d,p) level of theory. The Förster theory of energy transfer was considered to assess the energy transfer ability of the heterostructures [ 15 ]. The FRET rate formula has been derived analytically for a donor with localized states and an acceptor with 2D-confined states [ 20 , 26 ]: $$\:{\gamma\:}_{F}\left(R\right)=\underset{0}{\overset{\frac{{E}_{g}}{\hslash\:{\nu\:}_{F}}}{\int\:}}\frac{{e}_{0}^{2}}{96\pi\:\hslash\:{\epsilon\:}_{0}^{2}}({d}_{x}^{2}+{d}_{y}^{2}+{d}_{z}^{2})\frac{{e}^{-2qR}{q}^{3}}{\sqrt{{E}_{g}^{2}-{\nu\:}_{F}^{2}{q}^{2}}}dq$$ 6 , where ν F is the Fermi velocity near the Dirac point of Gr obtained from our calculations, E g is the HOMO-LUMO gap of an adsorbed NG and d i are the components of the molecular transition electric dipole moment of the lowest-energy excited state: \(\:{d}_{M}=\left({d}_{x},{d}_{y},{d}_{z}\right)=-{e}_{0}\int\:dr{{\Phi\:}}_{M}^{l*}r{{\Phi\:}}_{M}^{h}.\) (7). Energy gaps E g and transition dipole moments d M were computed using linear response time-dependent TD-DFT calculations implemented in Gaussian16, using the HSE06 functional [ 39 ] along with the 6-31G(d,p) Pople’s basis set. Natural transition orbitals (NTOs) [ 48 ] were computed and rendered to display the location of electron and holes in the relevant bright excited states. Results and discussion We have analyzed a total of six different NG quantum dots characterized primarily by the size of the NG fragment and its symmetry. Five of the selected model structures (C 42 , C 54 , C 60 , C 78 , C 96C ) were identical to the ones analyzed in the study of Wang et al. [ 8 ], comprising a set of centrosymmetric and elongated NGs. An additional structure comprising 96 carbon atoms in an elongated geometry (C 96L ) has been added. Altogether, the calculated molecules can be divided into two groups depending on the symmetry order: centrosymmetric molecules with a perpendicular C 3 axis C 42 and C 96C , and elongated molecules with only an in-plane C 2 axis C 54 , C 60 , C 78 , and C 96L . In the paper of Wang et al [ 8 ], it was reported that an increase of NG size from 42 to 96 carbon atoms enhances the NG/Gr interfacial coupling strength and increases CT efficiency, but the CT rate was not exactly quantified or discussed in relation to other competing processes. Our results from ground state calculations demonstrate that with the increasing number of carbon atoms/carbon cores, the interaction energy acting between NGs and Gr is increased (Fig. 3 a). These interactions are governed by dispersion, and the dispersion energy also increases with the increasing size of the NG. Additionally, we observed that NGs with larger lateral sizes adhered much stronger to Gr due to the enhanced interfacial vdW interactions, even when normalized to the number of carbon atoms (Fig. 3 b). These values indicate a logarithmic relationship between the dispersion energy and the number of carbon cores, implying that as the size of the NG increases, a point will be reached where further growth no longer enhances the interfacial vdW interactions, but it reaches its maximum value. Analyzing the electronic properties of the NG/Gr interfaces, we observed (as expected) that the energy difference between the highest occupied molecular orbital (HOMO) of the NGs and the Fermi energy level of Gr decreases with the increase in NG size from − 0.88, − 0.67, − 0.62, − 0.49, -0.43, − 0.59 eV for C 42 , C 54 , C 60 , C 78 , C 96L and C 96C (Fig. 3 c), respectively, due to an energy destabilization of the HOMO (Fig. 3 c) level of the NGs. Interestingly, the symmetry affects the driving force for hole transfer in the ground state as the decrease is not linear, i.e., the difference of HOMO of NG and the Fermi level of Gr has a value of -0.43 eV for C 96L , compared to the − 0.59 eV of its higher-symmetry counterpart C 96C , despite having similar values of interaction/dispersion energy. From the ground state analysis of the charge density difference (CDD) maps obtained via Bader analysis it is shown that before photoexcitation, a positive partial charge is localized on the NG and negative partial charge on Gr. Interestingly, the amount of charge transferred increases when the size of NGs is increased (Fig. 4 ). Positive density is located not only on NGs, but also on Gr at the regions below the hydrogenated edges of NGs (blue regions in Fig. 4 (i-vi)). Furthermore, the sum of net atomic charges is positive for NGs and negative for Gr in all studied systems. With the increasing size of NGs the positive/negative Bader charge linearly increases from 0.03 to 0.04 |e|, suggesting higher CT due to an increase in interaction energy. The increased number of holes on larger NG shall be reasoned by the above mentioned enhanced vdW interaction between NG and Gr in the heterojunction. Altogether, ground state calculations revealed that the size and symmetry of the NGs significantly influence the interfacial interactions within NG/Gr heterojunctions. As the size of the NGs increases, the vdW interaction strengthens, leading to an increased localization of positive charge on the NG part of the junction. Charge transfer The excited state analysis reveals that the lowest excited state of the parent NGs are red-shifted from C 42 < C 54 < C 60 < C 78 < C 96C < C 96L (Tables S1-S6), nevertheless, for C 42 , C 54 , C 60 and C 96C these S 1 states are dark. A closer look at the molecular orbitals involved in the electronic vertical transitions shows that HOMO→LUMO excitations in C 42 and C 96C NGs are forbidden due to their C 3 symmetry. In addition, the S 2 transition is characterized by HOMO→LUMO (50%) and HOMO-1→LUMO + 1 (49%) for both C 3 molecules and it is a dark state (Tables S1, S6). As such, the first bright excited state for C 42 is S 5 positioned at λ max = 358 nm (f = 0.724), which is the most blue-shifted among all studied NGs. For C 54 and C 60 , where the symmetry is decreased to C 2 , the lowest bright energy peaks are S 0 → S 2 with λ max = 459 nm (f = 0.724) and 487 nm (f = 0.257), respectively, and they are both characterized by the major contribution from HOMO→LUMO orbitals (Tables S2, S3). A further increase in length to obtain C 78 and C 96L lead to the bright S 0 → S 1 excitation located at λ max = 558 nm (f = 0.688) and 619 nm (f = 1.140) and, again, it is primarily a HOMO→LUMO excitation (Tables S4, S5). In contrast to C 96L , its C 3 counterpart C 96C has S 3 as its first bright state with λ max = 488 nm (f = 1.263). The role of the symmetry is also present in the shape of the calculated absorption spectra (Fig. 5 a). While for C 3 molecules (C 42 and C 96C ) only one absorption peak is obtained, for C 2 molecules there are more absorption peaks across the visible part of the spectra, where a low intensity, broad and red-shifted peak appears. To compute the CT properties at these interfaces, semiclassical Marcus theory was employed (see Computational details section). For the photoinduced hole transfer mechanism (PHT), size-related trends of both λ and J parameters favor the increase of the overall rate. The calculated J h values gradually increase with the increasing size of the NGs, except for C 96L , where the value is around 100 meV smaller then C 60 , suggesting, again, the strong role played by symmetry in CT properties. The opposite trend of monotonic decrease of reorganization energy λ with increasing the size of the NGs, from 100 to 58 meV is also observed (see Table 1 ). Overall, the constant increase in hole coupling values together with the decrease of the reorganization energy, lead to a strong increase in hole transfer rate ( γ PHT ) obtained from the Marcus equation, with an increase in hole transfer rate from 0.79 to 3.05 fs − 1 for C 42 and C 96C . Interestingly, the breaking of the symmetry from C 42 to C 54 strongly inhibits the hole transfer rate, with a fast decrease from 0.80 to 0.50 fs − 1 . Moreover, increasing the length of the NGs while keeping the same C 2 symmetry, from C 54 to C 96L , steadily increases the hole transfer rate up to 0.83 fs − 1 . The effect of the symmetry is clearly visible for the C 96 heterojunctions, where 3.6 times increase in hole transfer is observed when the symmetry is increased from C 2 to C 3 . This is in exact agreement with what was observed experimentally in PHT measurements at the two interfaces [ 8 ]. The observation of a one order of magnitude increase in CT efficiency between the smallest and the largest NGs (Fig. 5 b) allows us to conclude that the analysis of ground state CDD alone is not robust enough to correctly predict CT properties and one needs to move to the description of the excited states. Although the computed J e values are similar to J h values in heterojunctions with centrosymmetric molecules C 42 /Gr and C 96C /Gr (Table 1 ), the hole transfer rate γ PHT is higher due to a higher reorganization energy for the photoinduced electron transfer (PET) process (Fig. 5 b, Table 1 ). Apart from these two centrosymmetric heterojunctions, the J values for electron coupling are more than one order of magnitude lower than for hole coupling (Table 1 ), resulting in higher γ PHT than γ PET for all the probed heterojunctions. Altogether, these effects point to 1) a central role of orbital coupling in establishing PHT as the dominant mechanism; 2) a combination of orbital coupling and interfacial energetics determining the size enhancement of PHT rates, and 3) the strong effect that symmetry has on the coupling values, resulting in higher hole transfer values for C 42 despite its small size, and C 96C . Table 1 Parameters governing the hole and electron transfer processes: electronic coupling, reorganization energies, and CT rates for hole and electron transfer. Shaded rows denote NGs without C 3 symmetry. Junction J h λ h γ PHT J e λ e γ PET meV meV (fs − 1 ) meV meV (fs − 1 ) C 42 /Gr 199.74 100.86 0.79 205.89 134.56 0.53 C 54 /Gr 142.30 86.70 0.50 41.62 106.57 0.03 C 60 /Gr 154.05 82.41 0.62 43.05 101.37 0.04 C 78 /Gr 223.42 72.83 1.53 54.35 85.04 0.07 C 96L /Gr 153.99 65.10 0.83 35.21 73.16 0.04 C 96C /Gr 278.39 58.43 3.05 258.11 71.79 2.08 Energy transfer The energy transfer process can also play a crucial role, as it can compete with CT, potentially making certain NGs more or less favorable for efficient CT. Therefore, studying the energy transfer is essential for a comprehensive understanding of the overall performance of these heterojunctions. By evaluating Eq. ( 6 ) at equilibrium distances R 0 , we obtain high transfer rates γ F (R 0 ) for all NGs. They are determined by the square of the transition dipole moment d 2 of the relevant excited state, the electronic HOMO-LUMO gap of the NG donor E g , and the Fermi velocity ν F near the graphene Dirac point. The FRET rate is directly proportional to d 2 =(d x 2 +d y 2 +2d z 2 ) and shows an optimal value with respect to E g /ν F . From our calculations we obtain a value of ν F = 0.8·10 − 6 ms − 1 for all NG/Gr heterojunctions, yielding an optimal energy gap value of 3.20 eV. This value is very close to the calculated band gap of C 42 (Table 2 ). As the NG fragment is enlarged, and the symmetry is broken, the energy gap closes, reaching a minimum of 1.87 eV for C 96L . Once more, assuming a constant value for the transition dipole moment, an energy gap of 1.87 eV would yield a FRET rate 74% of the optimal value (Fig. 6 ). Within the C 3 and C 2 NGs, we observe an increase of d 2 values with the increasing number of atoms in the NGs. In C 3 compounds, the transition dipole moment is isotropic in the xy plane, whereas for the lower C 2 symmetry, it is oriented along the elongation axis (d x ) of C 54 , C 60 , C 78 , C 96L (see Table 2 ). For the largest NGs with 96 carbon atoms, d 2 is higher when the C 3 symmetry is obeyed, once more confirming the role that symmetry plays also for the FRET process. Despite having the least optimal gap (as discussed above), owing to a high transition dipole moment value, C 96L achieves the second-highest energy transfer rate, surpassed only by its C 3 -symmetric counterpart with the same number of atoms, C 96C . The FRET rate is greatly influenced on the equilibrium distance R 0 between the Gr plane and the NG fragment. R 0 is determined by π-π interaction between the aromatic rings and changes are negligible with the increase in NG size. Nonetheless, in the proximity of R 0 , the dependence of the FRET rate is particularly sensitive to small changes in distance. At these low distances (below R ≈ 5 Å), FRET rate γ F (R) shows exponential behavior and quickly decays as distance is increased, and is approximated by an exponential function γ F (R) ≈ α 1 e −α2R . At R = 5 Å, the distance dependence shifts from exponential to R − 4 decay, approximated by the power function γ F (R) ≈ α 3 R −4 , characteristic of a dipole with localized energy levels interacting with a plane with delocalized states [ 20 ]. The obtained fitted decay parameters α 1 , α 2 , α 3 are listed in Table 1 , whereas the exact and fitted FRET curves are plotted in Fig. 7 a. The visualization facilitates the discussion of crossover between the exponential regime (displayed as blue dashed curves in Fig. 7 a) and the R − 4 regime (red dashed-dotted curves). The optimized distance R 0 at which the FRET rate is compared (black circles) is always below the crossover point. However, the fitting obtained for C 42 places it the closest to the intersection, while for C 96L it is the furthest from the crossover point. Simultaneously, at the highest distance of 10 Å, the C 96L exact curve (thick black line) is still relatively close to the exponential regime, while for smaller NGs it is already firmly following the power law. This shows that the crossover distance cannot be easily generalized as it clearly depends on the molecular size and symmetry. The exponential decay rates α 1 are similar for C 42 and C 96C , with values up to ten times higher than for the rest of NGs possessing C 2 symmetry, in which case α 1 slowly increases with system size from 1.34 to 4.55 fs − 1 (Table 2 ). α 2 is the highest for the smallest C 42 system and decreases with system size with the exception of a small increase in C 96C . However, all values are in the 0.6-1 Å −1 range, hence we can assume that this parameter is not the crucial one in determining the FRET rate. The long-range decay constant α 3 shows the widest range of values, reaching up to 260 fs − 1 for the largest centrosymmetric NG. This softening of the long-range decay can be interpreted considering the extended delocalization of the C 96C NG and shifting from the R − 4 dependence towards the R − 2 which appears between two fully delocalized sheets, i.e. a Gr/Gr interface. Table 2 Parameters governing the Förster energy transfer process: energy gaps, transition dipole moment components, oscillator strengths of the transitions, DFT-optimized distance from the Gr plane, FRET rates, and decay parameters for calculated NG systems adsorbed on Gr. Shaded rows denote NGs without C 3 symmetry. Junction E g d x d y d z d 2 R 0 γ F (R 0 ) α 1 α 2 α 3 (eV) (e 0 a 0 ) (e 0 a 0 ) (e 0 a 0 ) (e 0 a 0 ) 2 (Å) (fs − 1 ) (fs − 1 ) (Å −1 ) (fs − 1 ) C 42 /Gr 3.15 2.90 2.89 0.00 16.75 3.248 0.83 20.75 1.02 98.36 C 54 /Gr 2.65 -1.55 0.03 0.00 2.41 3.262 0.11 1.78 0.86 17.39 C 60 /Gr 2.48 -2.03 -0.01 0.00 4.12 3.291 0.18 2.50 0.81 32.28 C 78 /Gr 2.10 3.55 0.00 0.00 12.63 3.305 0.50 4.65 0.68 122.60 C 96L /Gr 1.87 4.82 -0.01 0.00 33.12 3.381 1.11 8.64 0.61 372.26 C 96C /Gr 2.25 4.42 -4.41 0.00 38.96 3.295 1.62 17.49 0.73 347.47 Table S2 Excited state properties of the parent C 54 NG. Excited state Wavelength (nm) Osc. Strength (a.u.) E (eV) Transitions dS 2 (a.u.) S 1 468.8 0.000 2.6 H-1→LUMO (50%), HOMO→L + 1 (48%) 0.004 S 2 459.3 0.159 2.7 H-1→L + 1 (14%), HOMO→LUMO (83%) 2.407 S 3 406.9 0.034 3.0 H-2→LUMO (20%), HOMO→L + 1 (12%), HOMO→L + 2 (63%) 0.459 S 4 391.0 0.495 3.2 H-1→L + 1 (78%), HOMO→LUMO (11%) 6.366 S 5 389.3 0.528 3.2 H-2→LUMO (17%), H-1→LUMO (40%), HOMO→L + 1 (35%) 6.771 S 6 376.5 0.139 3.3 H-2→L + 1 (23%), H-1→L + 2 (67%) 1.719 S 7 365.0 0.269 3.4 H-2→LUMO (59%), HOMO→L + 2 (29%) 3.231 S 8 351.3 0.113 3.5 H-2→L + 1 (65%), H-1→L + 2 (19%) 1.301 S 9 346.4 0.014 3.6 HOMO→L + 3 (93%) 0.154 S 10 342.5 0.001 3.6 HOMO→L + 4 (85%) 0.014 Energy transfer shows a similar size- and symmetry-related trends to charge transfer. The high-symmetry NGs C42 and C96C show high transfer rates for all three considered processes: electron, hole, and energy transfers. To disentangle the effect of symmetry on the charge and energy transfer processes, we will focus the discussion on the two prevalent processes in NG/Gr nanojunctions: PHT and FRET. All of the calculated charge and energy transfer rates are on the order of 10 13 -10 15 s − 1 . The most prevalent CT process is the hole transfer, which occurs at the highest rate for every considered system. This is in agreement with experiment, in which the PHT prevails in NG/Gr heterostructures [ 8 ]. PET is predicted to be significant only in centrosymmetric NGs, and negligible when the symmetry is broken. FRET rate exceeds both CT rates for the smallest centrosymmetric C 42 /Gr heterojunction. For longer C 2 systems its rate is decreased. However, unlike PET, steadily increase with NG size, reaching the highest value for the C96 C /Gr interface. In both cases of PHT and FRET, the rates are high for C 3 -symmetric NGs, and strongly decrease for C 2 compounds. In case of PHT, this symmetry-driven deterioration is overcome when the size is increased from 42 to 78 atoms, shown for C 78 by γ PHT (C 78 /Gr) > γ F (C 42 /Gr), and similarly for larger NGs. For FRET, none of the calculated C 2 -symmetric NGs exceed the transfer rate of C 42 . Therefore, the additional symmetry appears to play a bigger role in case of PHT than FRET. This tells us that if a junction with quenched energy transfer is desirable, then the use of a larger NG may be beneficial. Summary and conclusions We identified several characteristics of NGs that influence the transfer rates: the HOMO-LUMO gap, excited state properties, governed to a degree by the lateral size and symmetry of the NG fragment. We were also able to discern the graphene Fermi velocity near the Dirac point as a parameter influencing the optimal energy transfer rate. With the possibility of Fermi velocity engineering by doping, strain, or applying an external potential [ 49 , 50 ], also the Gr part of the nanojunction can be tailored, if needed, to an application with a specific NG. The study provides a thorough analysis of the observed trends, including the impact of parameters appearing in the rate equations for all three processes. The current computational study can serve as a guideline as to the achievable properties and limits of these and similar systems. The synthesis of large NGs, with up to at least 132 atoms, over 30 Å lateral size, and beyond can be achieved [ 4 ], with a number of geometries going beyond the C 3 and elongated C 2 shapes analyzed here. Declarations Author Contribution Conceptualization, S.O.; methodology, O. S. and S.O.; validation, formal analysis and investigation M. W. and M. L.; resources,M. W., M. L., and S.O.; data curation, S.O.; writing—original draft preparation, M. W., M. L.. and S.O.; writing—reviewand editing, M. W., M. L. O. S. and S.O.; supervision, S.O.; project administration, S.O.; funding acquisition, S.O. All authors have read and agreed to the published version of the manuscript. Acknowledgments We gratefully acknowledge Polish high-performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2023/016865 and for awarding this project access to the LUMI supercomputer, owned by the EuroHPC Joint Undertaking, hosted by CSC (Finland) and the LUMI consortium through PLL/2023/05/016760. This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90254). M.L. acknowledges that this article has been produced with the financial support of the European Union under the REFRESH – Research Excellence For REgion Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048 via the Operational Programme Just Transition. S. O. is grateful to the National Science Centre, Poland for funding (grant no. UMO/2020/39/I/ST4/01446 and UMO-2023/50/E/ST4/00197). This research was carried out with the support of the Interdisciplinary Center for Mathematical and Computational Modeling at the University of Warsaw (ICM UW) under grants no. G83-28 and GB80-24. References X. Yan, B. Li, L.S. Li, Colloidal graphene quantum dots with well-defined structures, Acc Chem Res 46 (2013) 2254–2262. C. Moreno, M. Vilas-Varela, B. Kretz, A. Garcia-Lekue, M.V. Costache, M. Paradinas, M. Panighel, G. Ceballos, S.O. Valenzuela, D. Pena, A. Mugarza, Bottom-up synthesis of multifunctional nanoporous graphene, Science 360 (2018) 199–203. J. Liu, X. Feng, Synthetic Tailoring of Graphene Nanostructures with Zigzag-Edged Topologies: Progress and Perspectives, Angew Chem Int Ed Engl 59 (2020) 23386–23401. D. Medina-Lopez, T. Liu, S. Osella, H. Levy-Falk, N. Rolland, C. Elias, G. Huber, P. Ticku, L. Rondin, B. Jousselme, D. Beljonne, J.S. Lauret, S. Campidelli, Interplay of structure and photophysics of individualized rod-shaped graphene quantum dots with up to 132 sp(2) carbon atoms, Nat Commun 14 (2023) 4728. P. Fantuzzi, A. Candini, Q. Chen, X. Yao, T. Dumslaff, N. Mishra, C. Coletti, K. Müllen, A. Narita, M. Affronte, Color Sensitive Response of Graphene/Graphene Quantum Dot Phototransistors, The Journal of Physical Chemistry C 123 (2019) 26490–26497. Y. Yan, D. Zhai, Y. Liu, J. Gong, J. Chen, P. Zan, Z. Zeng, S. Li, W. Huang, P. Chen, van der Waals Heterojunction between a Bottom-Up Grown Doped Graphene Quantum Dot and Graphene for Photoelectrochemical Water Splitting, ACS Nano 14 (2020) 1185–1195. Z. Liu, H. Qiu, S. Fu, C. Wang, X. Yao, A.G. Dixon, S. Campidelli, E. Pavlica, G. Bratina, S. Zhao, L. Rondin, J.S. Lauret, A. Narita, M. Bonn, K. Mullen, A. Ciesielski, H.I. Wang, P. Samori, Solution-Processed Graphene-Nanographene van der Waals Heterostructures for Photodetectors with Efficient and Ultralong Charge Separation, J Am Chem Soc 143 (2021) 17109–17116. X. Yu, S. Fu, M. Mandal, X. Yao, Z. Liu, W. Zheng, P. Samori, A. Narita, K. Mullen, D. Andrienko, M. Bonn, H.I. Wang, Tuning interfacial charge transfer in atomically precise nanographene-graphene heterostructures by engineering van der Waals interactions, J Chem Phys 156 (2022) 074702. Z. Liu, S. Fu, X. Liu, A. Narita, P. Samori, M. Bonn, H.I. Wang, Small Size, Big Impact: Recent Progress in Bottom-Up Synthesized Nanographenes for Optoelectronic and Energy Applications, Adv Sci (Weinh) 9 (2022) e2106055. S. Cao, J. Wang, F. Ma, M. Sun, Charge-transfer channel in quantum dot-graphene hybrid materials, Nanotechnology 29 (2018) 145202. L.Y. Hsu, W. Ding, G.C. Schatz, Plasmon-Coupled Resonance Energy Transfer, J Phys Chem Lett 8 (2017) 2357–2367. S. Osella, M. Wang, E. Menna, T. Gatti, (INVITED) Lighting-up nanocarbons through hybridization: Optoelectronic properties and perspectives, Optical Materials: X 12 (2021). S. Qiao, M. Di, J.-X. Jiang, B.-H. Han, Conjugated porous polymers for photocatalysis: The road from catalytic mechanism, molecular structure to advanced applications, EnergyChem 4 (2022). M. Wang, M. Langer, R. Altieri, M. Crisci, S. Osella, T. Gatti, Two-Dimensional Layered Heterojunctions for Photoelectrocatalysis, ACS Nano 18 (2024) 9245–9284. T. Förster, Zwischenmolekulare Energiewanderung und Fluoreszenz, Annalen der Physik 437 (1948) 55–75. R.R. Chance, A. Prock, R. Silbey, Molecular Fluorescence and Energy Transfer Near Interfaces, in: I. Prigogine, S.A. Rice (Eds.), Advances in Chemical Physics, 1978, pp. 1–65. B.N.J. Persson, N.D. Lang, Electron-hole-pair quenching of excited states near a metal, Physical Review B 26 (1982) 5409–5415. Š. Kos, M. Achermann, V.I. Klimov, D.L. Smith, Different regimes of Förster-type energy transfer between an epitaxial quantum well and a proximal monolayer of semiconductor nanocrystals, Physical Review B 71 (2005). P.L. Hernández-Martínez, A.O. Govorov, Exciton energy transfer between nanoparticles and nanowires, Physical Review B 78 (2008). R.S. Swathi, K.L. Sebastian, Distance dependence of fluorescence resonance energy transfer, Journal of Chemical Sciences 121 (2009) 777–787. A. Campion, A.R. Gallo, C.B. Harris, H.J. Robota, P.M. Whitmore, Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances, Chemical Physics Letters 73 (1980) 447–450. A.P. Alivisatos, D.H. Waldeck, C.B. Harris, Nonclassical behavior of energy transfer from molecules to metal surfaces: Biacetyl(3nπ*)/Ag(111), The Journal of Chemical Physics 82 (1985) 541–547. J. Hill, S.Y. Heriot, O. Worsfold, T.H. Richardson, A.M. Fox, D.D.C. Bradley, Controlled Förster energy transfer in emissive polymer Langmuir-Blodgett structures, Physical Review B 69 (2004). E. Malic, H. Appel, O.T. Hofmann, A. Rubio, Forster-Induced Energy Transfer in Functionalized Graphene, J Phys Chem C Nanomater Interfaces 118 (2014) 9283–9289. E. Malic, H. Appel, O.T. Hofmann, A. Rubio, Energy-transfer in porphyrin‐ functionalized graphene, physica status solidi (b) 251 (2014) 2495–2498. O.Y. Semchuk, O.O. Havryliuk, A.A. Biliuk, Inductive-resonance energy transfer in hybrid carbon nanostructures, Himia, Fizika ta Tehnologia Poverhni 15 (2024) 328–339. R.A. Marcus, N. Sutin, Electron transfers in chemistry and biology, Biochimica et Biophysica Acta (BBA) - Reviews on Bioenergetics 811 (1985) 265–322. G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys Rev B Condens Matter 47 (1993) 558–561. G. Kresse, J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium, Phys Rev B Condens Matter 49 (1994) 14251–14269. G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational Materials Science 6 (1996) 15–50. G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys Rev B Condens Matter 54 (1996) 11169–11186. J.P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys Rev Lett 77 (1996) 3865–3868. G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Physical Review B 59 (1999) 1758–1775. S. Grimme, J. Antony, S. Ehrlich, H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J Chem Phys 132 (2010) 154104. S. Grimme, S. Ehrlich, L. Goerigk, Effect of the damping function in dispersion corrected density functional theory, J Comput Chem 32 (2011) 1456–1465. R.F.W. Bader, A Bond Path: A Universal Indicator of Bonded Interactions, The Journal of Physical Chemistry A 102 (1998) 7314–7323. G. Henkelman, A. Arnaldsson, H. Jónsson, A fast and robust algorithm for Bader decomposition of charge density, Computational Materials Science 36 (2006) 354–360. M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, G.A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A.V. Marenich, J. Bloino, B.G. Janesko, R. Gomperts, B. Mennucci, H.P. Hratchian, J.V. Ortiz, A.F. Izmaylov, J.L. Sonnenberg, Williams, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V.G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J.A. Montgomery Jr., J.E. Peralta, F. Ogliaro, M.J. Bearpark, J.J. Heyd, E.N. Brothers, K.N. Kudin, V.N. Staroverov, T.A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A.P. Rendell, J.C. Burant, S.S. Iyengar, J. Tomasi, M. Cossi, J.M. Millam, M. Klene, C. Adamo, R. Cammi, J.W. Ochterski, R.L. Martin, K. Morokuma, O. Farkas, J.B. Foresman, D.J. Fox, Gaussian 16 Rev. C.01, Wallingford, CT, 2016. A.V. Krukau, O.A. Vydrov, A.F. Izmaylov, G.E. Scuseria, Influence of the exchange screening parameter on the performance of screened hybrid functionals, J Chem Phys 125 (2006) 224106. B. Baumeier, J. Kirkpatrick, D. Andrienko, Density-functional based determination of intermolecular charge transfer properties for large-scale morphologies, Phys Chem Chem Phys 12 (2010) 11103–11113. H. Oberhofer, K. Reuter, J. Blumberger, Charge Transport in Molecular Materials: An Assessment of Computational Methods, Chem Rev 117 (2017) 10319–10357. C. Musumeci, S. Osella, L. Ferlauto, D. Niedzialek, L. Grisanti, S. Bonacchi, A. Jouaiti, S. Milita, A. Ciesielski, D. Beljonne, M.W. Hosseini, P. Samori, Influence of the supramolecular order on the electrical properties of 1D coordination polymers based materials, Nanoscale 8 (2016) 2386–2394. M. Dobbelin, A. Ciesielski, S. Haar, S. Osella, M. Bruna, A. Minoia, L. Grisanti, T. Mosciatti, F. Richard, E.A. Prasetyanto, L. De Cola, V. Palermo, R. Mazzaro, V. Morandi, R. Lazzaroni, A.C. Ferrari, D. Beljonne, P. Samori, Light-enhanced liquid-phase exfoliation and current photoswitching in graphene-azobenzene composites, Nat Commun 7 (2016) 11090. S. Osella, S. Knippenberg, Environmental effects on the charge transfer properties of Graphene quantum dot based interfaces, International Journal of Quantum Chemistry 119 (2018). S.F. Nelsen, S.C. Blackstock, Y. Kim, Estimation of inner shell Marcus terms for amino nitrogen compounds by molecular orbital calculations, Journal of the American Chemical Society 109 (1987) 677–682. J. Blumberger, Recent Advances in the Theory and Molecular Simulation of Biological Electron Transfer Reactions, Chem Rev 115 (2015) 11191–11238. M. Kaźmierczak, S. Giannini, S. Osella, Photoinduced energy and electron transfer at graphene quantum dot/azobenzene interfaces, Journal of Materials Chemistry C 12 (2024) 143–153. R.L. Martin, Natural transition orbitals, The Journal of Chemical Physics 118 (2003) 4775–4777. J.R.F. Lima, Controlling the energy gap of graphene by Fermi velocity engineering, Physics Letters A 379 (2015) 179–182. A. Diaz-Fernandez, L. Chico, J.W. Gonzalez, F. Dominguez-Adame, Tuning the Fermi velocity in Dirac materials with an electric field, Sci Rep 7 (2017) 8058. Additional Declarations No competing interests reported. Supplementary Files Supportinginformation.docx Cite Share Download PDF Status: Published Journal Publication published 27 Dec, 2024 Read the published version in Theoretical Chemistry Accounts → Version 1 posted Editorial decision: Revision requested 05 Nov, 2024 Reviews received at journal 05 Nov, 2024 Reviews received at journal 21 Oct, 2024 Reviewers agreed at journal 08 Oct, 2024 Reviewers agreed at journal 06 Oct, 2024 Reviewers invited by journal 06 Oct, 2024 Editor assigned by journal 03 Oct, 2024 Submission checks completed at journal 03 Oct, 2024 First submitted to journal 03 Oct, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5199549","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":374446187,"identity":"63c9cb33-80ce-4717-ada2-b4e18df3958a","order_by":0,"name":"Mateusz Wlazło","email":"","orcid":"","institution":"University of Warsaw","correspondingAuthor":false,"prefix":"","firstName":"Mateusz","middleName":"","lastName":"Wlazło","suffix":""},{"id":374446190,"identity":"5d6e5dd6-97f6-4d09-b994-04e75d411b45","order_by":1,"name":"Michal Langer","email":"","orcid":"","institution":"Technical University of Ostrava","correspondingAuthor":false,"prefix":"","firstName":"Michal","middleName":"","lastName":"Langer","suffix":""},{"id":374446191,"identity":"a56cb01e-9e79-42fa-9d1c-7428efc55058","order_by":2,"name":"Oleksandr Y. 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(b) Schematic depiction of the interaction between donor and acceptor point dipoles, a point dipole with a 2D-delocalized acceptor, and a 2D-delocalized donor with a 2D-delocalized acceptor.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/a68889c806be97b799f4d0a4.jpg"},{"id":69770178,"identity":"ede105f3-47de-48f0-a1c0-45a765cedf71","added_by":"auto","created_at":"2024-11-25 06:27:26","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":141174,"visible":true,"origin":"","legend":"\u003cp\u003eTop view on the model structures, where NGs (cyan – carbon; white – hydrogen) are adsorbed on the Gr surface (black).\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/f3fdc2c771ada47d8852494e.jpg"},{"id":69768951,"identity":"14992d12-f034-4505-b3ec-5dd13f05fc98","added_by":"auto","created_at":"2024-11-25 06:11:25","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":67102,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Interaction energy and (b) dispersion energy per carbon atom calculated for the six studied NG/Gr heterostructures, (c) density of states of hybrid systems projected on the Gr infinite honeycomb lattice (dashed black lines) and NG fragments (thick colored lines). The arrows indicate the energy difference between the graphene Fermi level (set at 0 eV, horizontal dotted line) and the NGs HOMO level.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/f3943979a73d89f689ae041e.jpg"},{"id":69770176,"identity":"2302c812-84ce-40e1-9ace-ce4c869d082e","added_by":"auto","created_at":"2024-11-25 06:27:25","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":152609,"visible":true,"origin":"","legend":"\u003cp\u003eCharge density difference maps showing the regions with (i-vi) positive (blue) and (vii-xii) negative (magenta) densities, which were evaluated as the charge density difference of a NG/Gr interface and the NG and Gr monomers. The numbers in the top panel refer to the total Bader charges accumulated on NG and Gr.\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/ca1804cdab0c40e2e8815382.jpg"},{"id":69769078,"identity":"b6f6958a-e376-4ae1-9c54-7dea5f7e0353","added_by":"auto","created_at":"2024-11-25 06:19:25","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":77526,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Simulated absorption spectrum of six isolated NGs computed with the TDDFT at the HSE06/6-31G(d,p) level of theory. (b) Rates of hole and electron transfer as a function of nanographene size. Full and striped markers indicate C\u003csub\u003e3\u003c/sub\u003e-symmetric and C\u003csub\u003e2\u003c/sub\u003e-symmetric NGs, respectively.\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/0a4a362ec6a93d0f322664b2.jpg"},{"id":69769081,"identity":"4abd0784-1bed-4f8b-bddd-ddec7b460686","added_by":"auto","created_at":"2024-11-25 06:19:25","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":45421,"visible":true,"origin":"","legend":"\u003cp\u003eRelative FRET rate dependence on the donor HOMO-LUMO gap, assuming a constant transition dipole moment. Points with gaps corresponding to different NGs are marked with black circles.\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/5e7854dbbb4436620aa61719.jpg"},{"id":69768954,"identity":"682bb9a2-4987-4c1e-8367-f24afd4c05a9","added_by":"auto","created_at":"2024-11-25 06:11:25","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":79806,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy transfer rate as a function of NG-Gr distance. The values of energy transfer rates at the optimized molecule-surface distance R\u003csub\u003e0\u003c/sub\u003e are marked with black circles. Individual curves were fitted to short-range exponential (dashed blue) and long-range R\u003csup\u003e-4\u003c/sup\u003e (dash-dotted red) decay.\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/bea49cd1cb92b91297cb2a44.jpg"},{"id":72641054,"identity":"436b89bd-3e74-47f5-b3f4-dd14c8b9751a","added_by":"auto","created_at":"2024-12-30 16:11:01","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1398909,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/b70981de-9dd7-4f12-9198-48c4781ed8eb.pdf"},{"id":69770560,"identity":"22d83304-e13f-4063-805c-73b85271bb1c","added_by":"auto","created_at":"2024-11-25 06:35:30","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":3485642,"visible":true,"origin":"","legend":"","description":"","filename":"Supportinginformation.docx","url":"https://assets-eu.researchsquare.com/files/rs-5199549/v1/8702485c33e4692027019a88.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Unraveling the competition between charge and energy transfer in 0D/2D nanographene-graphene heterojunctions","fulltext":[{"header":"Introduction","content":"\u003cp\u003eGraphene molecules, known as nanographenes (NGs), can be synthesized with increasingly controlled shapes and edges thanks to the use of the bottom-up approach [\u003cspan additionalcitationids=\"CR2 CR3\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Due to the possibility of exact shape and size control, intriguing size-dependent optoelectronic properties arise when interfaced with graphene (Gr) to form 0D/2D heterojunctions [\u003cspan additionalcitationids=\"CR6 CR7 CR8\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. To understand their function in devices such as quantum emitters [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], quantum dot transistors [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] and photodetectors [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], it is crucial to increase the knowledge of interface charge and energy transfer processes that occur at such heterojunctions. Herein, we extend the current knowledge of size-dependent charge transport properties [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] by providing additional insight from ab initio calculations, and introduce a completely new insight on the influence of NG size and symmetry on F\u0026ouml;rster resonant energy transfer (FRET), a photophysical process that is often considered to be in competition with charge transfer (CT) at low length scales of a few Angstroms [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], as is the case in vertical NG/Gr heterostructures.\u003c/p\u003e \u003cp\u003eThe nanographene/graphene (NG/Gr) interfaces have already been investigated experimentally due to their interesting light interaction properties [\u003cspan additionalcitationids=\"CR6 CR7 CR8\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. In a study case by Wang et al. [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], the photoinduced hole transfer (PHT) was identified as the dominant CT mechanism, reconciling two competing views on the dependence of CT rate on the NG size. They suggested that van der Waals (vdW) interactions play a dominant role in determining the interfacial CT efficiency, whereas interfacial energetics and reorganization energy have a minor influence. Owing to the size-dependent increase in the interfacial coupling strength, the efficiency increased by one order of magnitude despite the decrease in the driving force of the hole transfer process. However, the mechanism behind this behavior is still elusive, prompting the current investigation via a detailed and systematic computational approach.\u003c/p\u003e \u003cp\u003eThe standard computational approach for estimation of charge or energy transfer prevalence at a heterojunction, would be based on the energy alignment of frontier orbitals. Depending on this molecular orbitals\u0026rsquo; level alignment, two different types of junctions can be obtained, namely Type-I where the energy transfer process occur, and Type-II heterojunctions, where the charge transfer mechanism prevails. Nonetheless, in the current case where 0D heterojunctions are physisorbed on a 2D graphene monolayer surface, is not clear to foresee a priori if the prevailing process is charge or energy transfer, as the quantum confinement effect, different charge carrier localization, exciton dynamics, etc., leads to complex interfacial interactions that are not easily captured by the traditional Type-I or Type-II classifications (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) [\u003cspan additionalcitationids=\"CR13\" citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eTo further develop the understanding of size-dependent vdW interactions in NG/Gr systems, we computationally explore the CT and FRET mechanisms (see Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea) applied to the same set of NGs. The results shed light on whether FRET efficiency follows a similar trend as CT and contribute to the understanding of the competition between these two mechanisms.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFRET is a non-radiative process of transferring energy from an excited donor species to an acceptor [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. When two fluorescent molecules are considered, the interface can be modelled as a pair of interacting dipoles. This leads to a R\u003csup\u003e-6\u003c/sup\u003e dependence, characteristic for all dipole-dipole interactions which take a major part in non-covalent bonding. Differences arise when low dimensional materials are considered. In fact, now a molecular dipole (a 0D material) interacting with a surface that can be approximated as an infinite plane, such as a molecule adsorbed on a Gr sheet (i.e., a material with 2D charge delocalization), differs quantitively from the simple dipolar approximation. The difference is due to the change of dimensionality of the acceptor states. The R\u003csup\u003e-6\u003c/sup\u003e dependence is observed when the excited states of both the donor and acceptor are localized, and thus the dipolar approximation is valid. Deviations occur when the donor, or both the donor and the acceptor, have delocalized charge densities (see Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb). In these cases, the R\u003csup\u003e-6\u003c/sup\u003e power law softens depending on the carrier confinement dimensionality.\u003c/p\u003e \u003cp\u003eNumerous cases of different delocalized interfaces have been derived theoretically [\u003cspan additionalcitationids=\"CR17 CR18 CR19\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] and observed experimentally [\u003cspan additionalcitationids=\"CR22\" citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e], confirming the predicted deviations from the R\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e law (0D/0D). For instance, a nanoparticle donor-nanowire acceptor (0D/1D) system was calculated to follow a R\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e power law [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], a molecule on a bulk metal slab (0D/3D) a R\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e dependence on distance [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] (verified experimentally for pyrazine on Ni(111) [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]), and two fluorescent polymer films (PFO and F8BT) transfer the energy between each other with a R\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e dependence [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. The derivations and experiments that are most relevant to the current case of NGs on Gr and result in a R\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e dependence include excited molecular donors coupled to surface states in a metal [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] (verified e.g. by experiment with biacetyl molecule on silver [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]), a quantum well coupled to a quasicontinuum of high-energy quantum dot states [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] (verified in a InGaN QW/CdSe QD device), and, most recently, a fluorescent molecule on Gr [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. Swathi et al. [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] were able to derive the localized donor/graphene energy transfer rate formula analytically within the FRET effect [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. A few computational applications of this derivation have revealed that in molecular donor/graphene heterojunctions, the competing Dexter energy transfer mechanism is negligible compared to FRET [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e], which is the focus of current work.\u003c/p\u003e \u003cp\u003eIn the current study, we combine the analytical approaches for FRET [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e] and Marcus [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e] for CT with ab initio computations. We aim to assess the strength of both energy and charge transfer mechanisms at a 0D/2D interfaces consisting of planar NGs with different sizes and symmetries physisorbed on a Gr monolayer (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). A research question that motivates this work is to see the effect of delocalization length and change in symmetry on both CT and FRET properties, and the effect of the donor size on the interface, as it can behave less like localized donor/Gr (0D/2D) and increasingly like two delocalized planes (2D/2D), reminiscent of the interaction of two fluorescent polymer films that was shown to exhibit a R\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e distance dependence rather than R\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. Deriving a size limit in which delocalized interaction dominates over the localized limit will support the experimental efforts related to tunable-by-size NG optoelectronic properties.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"Computational details","content":"\u003cp\u003eA 12\u0026times;7 rectangular Gr supercell (30\u0026times;30 \u0026Aring; in the xy plane) consisting of 336 carbon atoms was generated, and each one of the six NGs considered in this study were inserted above the graphene plane to create the heterojunction. Full geometry optimizations of the atomic coordinates of the six model heterojunctions (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) were performed with first-principles calculations using planewave DFT implemented in VASP [\u003cspan additionalcitationids=\"CR29 CR30\" citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. In a first optimization step, both ionic positions and cell shape was allowed to relax, and in the final one, only the zcoordinates of ionic positions of the NG were allowed to be optimized until forces acting on atoms were lower than the 0.02 eV/\u0026Aring; threshold. A vacuum layer of 15 \u0026Aring; was considered along the z-direction to avoid spurious interactions between the images of the (nano)graphene layers due to the periodic boundary conditions. The electronic exchange and correlation effects were treated by the Perdew\u0026thinsp;\u0026minus;\u0026thinsp;Burke\u0026thinsp;\u0026minus;\u0026thinsp;Ernzerhof (PBE) functional [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e] with PAW-type pseudopotentials [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]. PBE-D3(BJ) theory level was applied to take into account the vdW interactions [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e]. The planewave basis set cutoff of 500 eV was used, and Brillouin zone integration scheme on a 2\u0026times;2\u0026times;1 k-point grid. For the charge density difference and Bader charge analysis [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e], calculations were carried out on a denser 4\u0026times;4\u0026times;1 k-point grid.\u003c/p\u003e \u003cp\u003eBader charge analysis was performed to evaluate the charge located on each atom (q\u003csub\u003eBader\u003c/sub\u003e)\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{q}_{Bader}={V}_{Val}-{N}_{Bader}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(1),\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere V\u003csub\u003eval\u003c/sub\u003e and N\u003csub\u003eBader\u003c/sub\u003e denote the number of valence electrons in a free atom and the computed number of valence electrons in the atom in the system, respectively.\u003c/p\u003e \u003cp\u003eThe interaction/dispersion energies \u003cem\u003eE\u003c/em\u003e\u003csub\u003eint\u003c/sub\u003e, \u003cem\u003eE\u003c/em\u003e\u003csub\u003edisp\u003c/sub\u003e acting between NG and Gr were evaluated as\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{int/disp}={E}_{AB}\\left(AB\\right)-{E}_{AB}\\left(A\\right)-{E}_{AB}\\left(B\\right),\\)\u003c/span\u003e\u003c/span\u003e (2),\u003c/p\u003e \u003cp\u003ewhere E(AB) is the total energy of the interface and E(A) and E(B) the energy of the isolated fragments with the same geometry as in the interface. Charge and energy transfer properties were derived from ground and excited state calculations implemented in Gaussian16 [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e], at the DFT HSE06/6-31G(d,p) [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e] functional and basis set level of theory. In order to quantify the CT rate at the interfaces, we adopt here the well-established semiclassical Marcus rate equation. In the weak coupling classical limits, the rate constant (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{if}\\)\u003c/span\u003e\u003c/span\u003e) for CT between two states (referred to as initial and final states, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|i⟩\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|f⟩\\)\u003c/span\u003e\u003c/span\u003e) can be written as:\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{if}=\\frac{2\\pi\\:}{ħ}{J}_{if}^{2}\\frac{1}{\\sqrt{4\\pi\\:\\lambda\\:{k}_{B}T}}\\text{e}\\text{x}\\text{p}(-\\frac{\\lambda\\:}{4\\lambda\\:{k}_{B}T})\\)\u003c/span\u003e\u003c/span\u003e, (3),\u003c/p\u003e \u003cp\u003e\u0026#119908;here \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{J}_{if}\\)\u003c/span\u003e\u003c/span\u003e is the electronic coupling between the electronic wave functions which is computed as described using the fragment orbital method [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e]. In this approach, the system is divided into fragments, in which an electron or hole is localized on a fragment and can hop from one fragment to another. The orbitals of a pair of molecules (a dimer) are projected onto a basis set defined by the orbitals of each individual molecule (the fragments). The obtained set of orthogonal molecular orbitals energies of the dimer are then used to rewrite the Fock matrix in the new localized basis set, to obtain a block-diagonal matrix. The main advantage of this method is the possibility to analyze pairs of different molecules, since it is not necessary to assume that the energies of the HOMO of the two fragments are equal, allowing for coupling from any occupied orbitals to any unoccupied ones [\u003cspan additionalcitationids=\"CR43\" citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e]. When degenerated states are present in the system, the procedure is modified to consider the effective coupling (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{J}_{eff}\\)\u003c/span\u003e\u003c/span\u003e) computed as the square root of the sum of the squared couplings related to the number of degenerate states found in the calculation. For example, for two degenerate initial states i1, i2 and two degenerate final states f1, f2, the effective coupling is\u003c/p\u003e\u003c/div\u003e \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{J}_{eff}=\\sqrt{{J}_{i1f1}^{2}+{J}_{i1f2}^{2}+{J}_{i2f1}^{2}+{J}_{i2f2}^{2}}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e.\u003c/p\u003e \u003cp\u003eThe reorganization energy λ is the energy cost to bring the nuclear configuration of the initial state into that of the final state while keeping the electronic configuration fixed at the initial state and was computed via the so-called four-point scheme (adiabatic potential energy surface method) [\u003cspan additionalcitationids=\"CR46\" citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e]:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}_{\\text{P}\\text{H}\\text{T}\\:}=\\:{E}^{+/-}\\left({Q}_{N}\\right)-{E}^{+/-}\\left({Q}^{+/-}\\right)+{E}_{N}\\left({Q}^{+/-}\\right)-{E}_{N}\\left({Q}_{N}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{N}\\left({Q}_{N}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}^{+/-}\\left({Q}^{+/-}\\right)\\)\u003c/span\u003e\u003c/span\u003e are the total energy of the optimized neutral and charged structures, while \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}^{+/-}\\left({Q}_{N}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{E}_{N}\\left({Q}^{+/-}\\right)\\)\u003c/span\u003e\u003c/span\u003e are the total energy after the vertical ionizations. The coupling and reorganization energy calculations were performed at the HSE06/6-31G(d,p) level of theory.\u003c/p\u003e \u003cp\u003eThe F\u0026ouml;rster theory of energy transfer was considered to assess the energy transfer ability of the heterostructures [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. The FRET rate formula has been derived analytically for a donor with localized states and an acceptor with 2D-confined states [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{\\gamma\\:}_{F}\\left(R\\right)=\\underset{0}{\\overset{\\frac{{E}_{g}}{\\hslash\\:{\\nu\\:}_{F}}}{\\int\\:}}\\frac{{e}_{0}^{2}}{96\\pi\\:\\hslash\\:{\\epsilon\\:}_{0}^{2}}({d}_{x}^{2}+{d}_{y}^{2}+{d}_{z}^{2})\\frac{{e}^{-2qR}{q}^{3}}{\\sqrt{{E}_{g}^{2}-{\\nu\\:}_{F}^{2}{q}^{2}}}dq$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere ν\u003csub\u003eF\u003c/sub\u003e is the Fermi velocity near the Dirac point of Gr obtained from our calculations, E\u003csub\u003eg\u003c/sub\u003e is the HOMO-LUMO gap of an adsorbed NG and d\u003csub\u003ei\u003c/sub\u003e are the components of the molecular transition electric dipole moment of the lowest-energy excited state:\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabd\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{d}_{M}=\\left({d}_{x},{d}_{y},{d}_{z}\\right)=-{e}_{0}\\int\\:dr{{\\Phi\\:}}_{M}^{l*}r{{\\Phi\\:}}_{M}^{h}.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(7).\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eEnergy gaps E\u003csub\u003eg\u003c/sub\u003e and transition dipole moments d\u003csub\u003eM\u003c/sub\u003e were computed using linear response time-dependent TD-DFT calculations implemented in Gaussian16, using the HSE06 functional [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e] along with the 6-31G(d,p) Pople\u0026rsquo;s basis set. Natural transition orbitals (NTOs) [\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e] were computed and rendered to display the location of electron and holes in the relevant bright excited states.\u003c/p\u003e"},{"header":"Results and discussion","content":"\u003cp\u003eWe have analyzed a total of six different NG quantum dots characterized primarily by the size of the NG fragment and its symmetry. Five of the selected model structures (C\u003csub\u003e42\u003c/sub\u003e, C\u003csub\u003e54\u003c/sub\u003e, C\u003csub\u003e60\u003c/sub\u003e, C\u003csub\u003e78\u003c/sub\u003e, C\u003csub\u003e96C\u003c/sub\u003e) were identical to the ones analyzed in the study of Wang et al. [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], comprising a set of centrosymmetric and elongated NGs. An additional structure comprising 96 carbon atoms in an elongated geometry (C\u003csub\u003e96L\u003c/sub\u003e) has been added. Altogether, the calculated molecules can be divided into two groups depending on the symmetry order: centrosymmetric molecules with a perpendicular C\u003csub\u003e3\u003c/sub\u003e axis C\u003csub\u003e42\u003c/sub\u003e and C\u003csub\u003e96C\u003c/sub\u003e, and elongated molecules with only an in-plane C\u003csub\u003e2\u003c/sub\u003e axis C\u003csub\u003e54\u003c/sub\u003e, C\u003csub\u003e60\u003c/sub\u003e, C\u003csub\u003e78\u003c/sub\u003e, and C\u003csub\u003e96L\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003eIn the paper of Wang et al [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], it was reported that an increase of NG size from 42 to 96 carbon atoms enhances the NG/Gr interfacial coupling strength and increases CT efficiency, but the CT rate was not exactly quantified or discussed in relation to other competing processes. Our results from ground state calculations demonstrate that with the increasing number of carbon atoms/carbon cores, the interaction energy acting between NGs and Gr is increased (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea). These interactions are governed by dispersion, and the dispersion energy also increases with the increasing size of the NG. Additionally, we observed that NGs with larger lateral sizes adhered much stronger to Gr due to the enhanced interfacial vdW interactions, even when normalized to the number of carbon atoms (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb). These values indicate a logarithmic relationship between the dispersion energy and the number of carbon cores, implying that as the size of the NG increases, a point will be reached where further growth no longer enhances the interfacial vdW interactions, but it reaches its maximum value.\u003c/p\u003e \u003cp\u003eAnalyzing the electronic properties of the NG/Gr interfaces, we observed (as expected) that the energy difference between the highest occupied molecular orbital (HOMO) of the NGs and the Fermi energy level of Gr decreases with the increase in NG size from \u0026minus;\u0026thinsp;0.88, \u0026minus;\u0026thinsp;0.67, \u0026minus;\u0026thinsp;0.62, \u0026minus;\u0026thinsp;0.49, -0.43, \u0026minus;\u0026thinsp;0.59 eV for C\u003csub\u003e42\u003c/sub\u003e, C\u003csub\u003e54\u003c/sub\u003e, C\u003csub\u003e60\u003c/sub\u003e, C\u003csub\u003e78\u003c/sub\u003e, C\u003csub\u003e96L\u003c/sub\u003e and C\u003csub\u003e96C\u003c/sub\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec), respectively, due to an energy destabilization of the HOMO (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec) level of the NGs. Interestingly, the symmetry affects the driving force for hole transfer in the ground state as the decrease is not linear, i.e., the difference of HOMO of NG and the Fermi level of Gr has a value of -0.43 eV for C\u003csub\u003e96L\u003c/sub\u003e, compared to the \u0026minus;\u0026thinsp;0.59 eV of its higher-symmetry counterpart C\u003csub\u003e96C\u003c/sub\u003e, despite having similar values of interaction/dispersion energy.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFrom the ground state analysis of the charge density difference (CDD) maps obtained via Bader analysis it is shown that before photoexcitation, a positive partial charge is localized on the NG and negative partial charge on Gr. Interestingly, the amount of charge transferred increases when the size of NGs is increased (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e). Positive density is located not only on NGs, but also on Gr at the regions below the hydrogenated edges of NGs (blue regions in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(i-vi)). Furthermore, the sum of net atomic charges is positive for NGs and negative for Gr in all studied systems. With the increasing size of NGs the positive/negative Bader charge linearly increases from 0.03 to 0.04 |e|, suggesting higher CT due to an increase in interaction energy. The increased number of holes on larger NG shall be reasoned by the above mentioned enhanced vdW interaction between NG and Gr in the heterojunction.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAltogether, ground state calculations revealed that the size and symmetry of the NGs significantly influence the interfacial interactions within NG/Gr heterojunctions. As the size of the NGs increases, the vdW interaction strengthens, leading to an increased localization of positive charge on the NG part of the junction.\u003c/p\u003e\n\u003ch3\u003eCharge transfer\u003c/h3\u003e\n\u003cp\u003eThe excited state analysis reveals that the lowest excited state of the parent NGs are red-shifted from C\u003csub\u003e42\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;C\u003csub\u003e54\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;C\u003csub\u003e60\u003c/sub\u003e \u0026lt; C\u003csub\u003e78\u003c/sub\u003e \u0026lt; C\u003csub\u003e96C\u003c/sub\u003e \u0026lt; C\u003csub\u003e96L\u003c/sub\u003e (Tables S1-S6), nevertheless, for C\u003csub\u003e42\u003c/sub\u003e, C\u003csub\u003e54\u003c/sub\u003e, C\u003csub\u003e60\u003c/sub\u003e and C\u003csub\u003e96C\u003c/sub\u003e these S\u003csub\u003e1\u003c/sub\u003e states are dark. A closer look at the molecular orbitals involved in the electronic vertical transitions shows that HOMO\u0026rarr;LUMO excitations in C\u003csub\u003e42\u003c/sub\u003e and C\u003csub\u003e96C\u003c/sub\u003e NGs are forbidden due to their C\u003csub\u003e3\u003c/sub\u003e symmetry. In addition, the S\u003csub\u003e2\u003c/sub\u003e transition is characterized by HOMO\u0026rarr;LUMO (50%) and HOMO-1\u0026rarr;LUMO\u0026thinsp;+\u0026thinsp;1 (49%) for both C\u003csub\u003e3\u003c/sub\u003e molecules and it is a dark state (Tables S1, S6). As such, the first bright excited state for C\u003csub\u003e42\u003c/sub\u003e is S\u003csub\u003e5\u003c/sub\u003e positioned at λ\u003csub\u003emax\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;358 nm (f\u0026thinsp;=\u0026thinsp;0.724), which is the most blue-shifted among all studied NGs. For C\u003csub\u003e54\u003c/sub\u003e and C\u003csub\u003e60\u003c/sub\u003e, where the symmetry is decreased to C\u003csub\u003e2\u003c/sub\u003e, the lowest bright energy peaks are S\u003csub\u003e0\u003c/sub\u003e \u0026rarr; S\u003csub\u003e2\u003c/sub\u003e with λ\u003csub\u003emax\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;459 nm (f\u0026thinsp;=\u0026thinsp;0.724) and 487 nm (f\u0026thinsp;=\u0026thinsp;0.257), respectively, and they are both characterized by the major contribution from HOMO\u0026rarr;LUMO orbitals (Tables S2, S3). A further increase in length to obtain C\u003csub\u003e78\u003c/sub\u003e and C\u003csub\u003e96L\u003c/sub\u003e lead to the bright S\u003csub\u003e0\u003c/sub\u003e \u0026rarr; S\u003csub\u003e1\u003c/sub\u003e excitation located at λ\u003csub\u003emax\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;558 nm (f\u0026thinsp;=\u0026thinsp;0.688) and 619 nm (f\u0026thinsp;=\u0026thinsp;1.140) and, again, it is primarily a HOMO\u0026rarr;LUMO excitation (Tables S4, S5). In contrast to C\u003csub\u003e96L\u003c/sub\u003e, its C\u003csub\u003e3\u003c/sub\u003e counterpart C\u003csub\u003e96C\u003c/sub\u003e has S\u003csub\u003e3\u003c/sub\u003e as its first bright state with λ\u003csub\u003emax\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;488 nm (f\u0026thinsp;=\u0026thinsp;1.263). The role of the symmetry is also present in the shape of the calculated absorption spectra (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea). While for C\u003csub\u003e3\u003c/sub\u003e molecules (C\u003csub\u003e42\u003c/sub\u003e and C\u003csub\u003e96C\u003c/sub\u003e) only one absorption peak is obtained, for C\u003csub\u003e2\u003c/sub\u003e molecules there are more absorption peaks across the visible part of the spectra, where a low intensity, broad and red-shifted peak appears.\u003c/p\u003e \u003cp\u003eTo compute the CT properties at these interfaces, semiclassical Marcus theory was employed (see \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003eComputational details\u003c/span\u003e section). For the photoinduced hole transfer mechanism (PHT), size-related trends of both λ and J parameters favor the increase of the overall rate. The calculated \u003cem\u003eJ\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e values gradually increase with the increasing size of the NGs, except for C\u003csub\u003e96L\u003c/sub\u003e, where the value is around 100 meV smaller then C\u003csub\u003e60\u003c/sub\u003e, suggesting, again, the strong role played by symmetry in CT properties. The opposite trend of monotonic decrease of reorganization energy λ with increasing the size of the NGs, from 100 to 58 meV is also observed (see Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Overall, the constant increase in hole coupling values together with the decrease of the reorganization energy, lead to a strong increase in hole transfer rate (\u003cb\u003eγ\u003c/b\u003e\u003csub\u003e\u003cb\u003ePHT\u003c/b\u003e\u003c/sub\u003e) obtained from the Marcus equation, with an increase in hole transfer rate from 0.79 to 3.05 fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e for C\u003csub\u003e42\u003c/sub\u003e and C\u003csub\u003e96C\u003c/sub\u003e. Interestingly, the breaking of the symmetry from C\u003csub\u003e42\u003c/sub\u003e to C\u003csub\u003e54\u003c/sub\u003e strongly inhibits the hole transfer rate, with a fast decrease from 0.80 to 0.50 fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. Moreover, increasing the length of the NGs while keeping the same C\u003csub\u003e2\u003c/sub\u003e symmetry, from C\u003csub\u003e54\u003c/sub\u003e to C\u003csub\u003e96L\u003c/sub\u003e, steadily increases the hole transfer rate up to 0.83 fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. The effect of the symmetry is clearly visible for the C\u003csub\u003e96\u003c/sub\u003e heterojunctions, where 3.6 times increase in hole transfer is observed when the symmetry is increased from C\u003csub\u003e2\u003c/sub\u003e to C\u003csub\u003e3\u003c/sub\u003e. This is in exact agreement with what was observed experimentally in PHT measurements at the two interfaces [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. The observation of a one order of magnitude increase in CT efficiency between the smallest and the largest NGs (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb) allows us to conclude that the analysis of ground state CDD alone is not robust enough to correctly predict CT properties and one needs to move to the description of the excited states.\u003c/p\u003e \u003cp\u003eAlthough the computed \u003cem\u003eJ\u003c/em\u003e\u003csub\u003ee\u003c/sub\u003e values are similar to \u003cem\u003eJ\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e values in heterojunctions with centrosymmetric molecules C\u003csub\u003e42\u003c/sub\u003e/Gr and C\u003csub\u003e96C\u003c/sub\u003e/Gr (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), the hole transfer rate \u003cem\u003eγ\u003c/em\u003e\u003csub\u003ePHT\u003c/sub\u003e is higher due to a higher reorganization energy for the photoinduced electron transfer (PET) process (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb, Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Apart from these two centrosymmetric heterojunctions, the \u003cem\u003eJ\u003c/em\u003e values for electron coupling are more than one order of magnitude lower than for hole coupling (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), resulting in higher \u003cem\u003eγ\u003c/em\u003e\u003csub\u003ePHT\u003c/sub\u003e than \u003cem\u003eγ\u003c/em\u003e\u003csub\u003ePET\u003c/sub\u003e for all the probed heterojunctions. Altogether, these effects point to 1) a central role of orbital coupling in establishing PHT as the dominant mechanism; 2) a combination of orbital coupling and interfacial energetics determining the size enhancement of PHT rates, and 3) the strong effect that symmetry has on the coupling values, resulting in higher hole transfer values for C\u003csub\u003e42\u003c/sub\u003e despite its small size, and C\u003csub\u003e96C\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters governing the hole and electron transfer processes: electronic coupling, reorganization energies, and CT rates for hole and electron transfer. Shaded rows denote NGs without C\u003csub\u003e3\u003c/sub\u003e symmetry.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eJunction\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eJ\u003csub\u003eh\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eλ\u003csub\u003eh\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eγ\u003csub\u003ePHT\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eJ\u003csub\u003ee\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eλ\u003csub\u003ee\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eγ\u003csub\u003ePET\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003emeV\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003emeV\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003emeV\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003emeV\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e(fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e42\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e199.74\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e100.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.79\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e205.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e134.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.53\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e54\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e142.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e86.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e41.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e106.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.03\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e60\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e154.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e82.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e43.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e101.37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.04\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e78\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e223.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e72.83\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e54.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e85.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.07\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e96L\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e153.99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e65.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.83\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e35.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e73.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.04\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e96C\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e278.39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e58.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e258.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e71.79\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.08\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e\n\u003ch3\u003eEnergy transfer\u003c/h3\u003e\n\u003cp\u003eThe energy transfer process can also play a crucial role, as it can compete with CT, potentially making certain NGs more or less favorable for efficient CT. Therefore, studying the energy transfer is essential for a comprehensive understanding of the overall performance of these heterojunctions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBy evaluating Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e6\u003c/span\u003e) at equilibrium distances R\u003csub\u003e0\u003c/sub\u003e, we obtain high transfer rates γ\u003csub\u003eF\u003c/sub\u003e(R\u003csub\u003e0\u003c/sub\u003e) for all NGs. They are determined by the square of the transition dipole moment d\u003csup\u003e2\u003c/sup\u003e of the relevant excited state, the electronic HOMO-LUMO gap of the NG donor E\u003csub\u003eg\u003c/sub\u003e, and the Fermi velocity ν\u003csub\u003eF\u003c/sub\u003e near the graphene Dirac point. The FRET rate is directly proportional to d\u003csup\u003e2\u003c/sup\u003e=(d\u003csub\u003ex\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e+d\u003csub\u003ey\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e+2d\u003csub\u003ez\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e) and shows an optimal value with respect to E\u003csub\u003eg\u003c/sub\u003e/ν\u003csub\u003eF\u003c/sub\u003e. From our calculations we obtain a value of ν\u003csub\u003eF\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.8\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e ms\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e for all NG/Gr heterojunctions, yielding an optimal energy gap value of 3.20 eV. This value is very close to the calculated band gap of C\u003csub\u003e42\u003c/sub\u003e (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). As the NG fragment is enlarged, and the symmetry is broken, the energy gap closes, reaching a minimum of 1.87 eV for C\u003csub\u003e96L\u003c/sub\u003e. Once more, assuming a constant value for the transition dipole moment, an energy gap of 1.87 eV would yield a FRET rate 74% of the optimal value (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWithin the C\u003csub\u003e3\u003c/sub\u003e and C\u003csub\u003e2\u003c/sub\u003e NGs, we observe an increase of d\u003csup\u003e2\u003c/sup\u003e values with the increasing number of atoms in the NGs. In C\u003csub\u003e3\u003c/sub\u003e compounds, the transition dipole moment is isotropic in the xy plane, whereas for the lower C\u003csub\u003e2\u003c/sub\u003e symmetry, it is oriented along the elongation axis (d\u003csub\u003ex\u003c/sub\u003e) of C\u003csub\u003e54\u003c/sub\u003e, C\u003csub\u003e60\u003c/sub\u003e, C\u003csub\u003e78\u003c/sub\u003e, C\u003csub\u003e96L\u003c/sub\u003e (see Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). For the largest NGs with 96 carbon atoms, d\u003csup\u003e2\u003c/sup\u003e is higher when the C\u003csub\u003e3\u003c/sub\u003e symmetry is obeyed, once more confirming the role that symmetry plays also for the FRET process. Despite having the least optimal gap (as discussed above), owing to a high transition dipole moment value, C\u003csub\u003e96L\u003c/sub\u003e achieves the second-highest energy transfer rate, surpassed only by its C\u003csub\u003e3\u003c/sub\u003e-symmetric counterpart with the same number of atoms, C\u003csub\u003e96C\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003eThe FRET rate is greatly influenced on the equilibrium distance R\u003csub\u003e0\u003c/sub\u003e between the Gr plane and the NG fragment. R\u003csub\u003e0\u003c/sub\u003e is determined by π-π interaction between the aromatic rings and changes are negligible with the increase in NG size. Nonetheless, in the proximity of R\u003csub\u003e0\u003c/sub\u003e, the dependence of the FRET rate is particularly sensitive to small changes in distance. At these low distances (below R\u0026thinsp;\u0026asymp;\u0026thinsp;5 \u0026Aring;), FRET rate γ\u003csub\u003eF\u003c/sub\u003e(R) shows exponential behavior and quickly decays as distance is increased, and is approximated by an exponential function γ\u003csub\u003eF\u003c/sub\u003e(R) \u0026asymp; α\u003csub\u003e1\u003c/sub\u003ee\u003csup\u003e\u0026minus;α2R\u003c/sup\u003e. At R\u0026thinsp;=\u0026thinsp;5 \u0026Aring;, the distance dependence shifts from exponential to R\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e decay, approximated by the power function γ\u003csub\u003eF\u003c/sub\u003e(R) \u0026asymp; α\u003csub\u003e3\u003c/sub\u003eR\u003csup\u003e\u0026minus;4\u003c/sup\u003e, characteristic of a dipole with localized energy levels interacting with a plane with delocalized states [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. The obtained fitted decay parameters α\u003csub\u003e1\u003c/sub\u003e, α\u003csub\u003e2\u003c/sub\u003e, α\u003csub\u003e3\u003c/sub\u003e are listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, whereas the exact and fitted FRET curves are plotted in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003ea.\u003c/p\u003e \u003cp\u003eThe visualization facilitates the discussion of crossover between the exponential regime (displayed as blue dashed curves in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003ea) and the R\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e regime (red dashed-dotted curves). The optimized distance R\u003csub\u003e0\u003c/sub\u003e at which the FRET rate is compared (black circles) is always below the crossover point. However, the fitting obtained for C\u003csub\u003e42\u003c/sub\u003e places it the closest to the intersection, while for C\u003csub\u003e96L\u003c/sub\u003e it is the furthest from the crossover point. Simultaneously, at the highest distance of 10 \u0026Aring;, the C\u003csub\u003e96L\u003c/sub\u003e exact curve (thick black line) is still relatively close to the exponential regime, while for smaller NGs it is already firmly following the power law. This shows that the crossover distance cannot be easily generalized as it clearly depends on the molecular size and symmetry.\u003c/p\u003e \u003cp\u003eThe exponential decay rates α\u003csub\u003e1\u003c/sub\u003e are similar for C\u003csub\u003e42\u003c/sub\u003e and C\u003csub\u003e96C\u003c/sub\u003e, with values up to ten times higher than for the rest of NGs possessing C\u003csub\u003e2\u003c/sub\u003e symmetry, in which case α\u003csub\u003e1\u003c/sub\u003e slowly increases with system size from 1.34 to 4.55 fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). α\u003csub\u003e2\u003c/sub\u003e is the highest for the smallest C\u003csub\u003e42\u003c/sub\u003e system and decreases with system size with the exception of a small increase in C\u003csub\u003e96C\u003c/sub\u003e. However, all values are in the 0.6-1 \u0026Aring;\u003csup\u003e\u0026minus;1\u003c/sup\u003e range, hence we can assume that this parameter is not the crucial one in determining the FRET rate. The long-range decay constant α\u003csub\u003e3\u003c/sub\u003e shows the widest range of values, reaching up to 260 fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e for the largest centrosymmetric NG. This softening of the long-range decay can be interpreted considering the extended delocalization of the C\u003csub\u003e96C\u003c/sub\u003e NG and shifting from the R\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e dependence towards the R\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e which appears between two fully delocalized sheets, i.e. a Gr/Gr interface.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters governing the F\u0026ouml;rster energy transfer process: energy gaps, transition dipole moment components, oscillator strengths of the transitions, DFT-optimized distance from the Gr plane, FRET rates, and decay parameters for calculated NG systems adsorbed on Gr. Shaded rows denote NGs without C\u003csub\u003e3\u003c/sub\u003e symmetry.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"11\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c11\" colnum=\"11\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eJunction\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eE\u003csub\u003eg\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ed\u003csub\u003ex\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ed\u003csub\u003ey\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003ed\u003csub\u003ez\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ed\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eR\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eγ\u003csub\u003eF\u003c/sub\u003e(R\u003csub\u003e0\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eα\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\"\u003e \u003cp\u003eα\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c11\"\u003e \u003cp\u003eα\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e(e\u003csub\u003e0\u003c/sub\u003ea\u003csub\u003e0\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(e\u003csub\u003e0\u003c/sub\u003ea\u003csub\u003e0\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e(e\u003csub\u003e0\u003c/sub\u003ea\u003csub\u003e0\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e(e\u003csub\u003e0\u003c/sub\u003ea\u003csub\u003e0\u003c/sub\u003e)\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e(\u0026Aring;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003e(fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003e(fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\"\u003e \u003cp\u003e(\u0026Aring;\u003csup\u003e\u0026minus;1\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c11\"\u003e \u003cp\u003e(fs\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e42\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e16.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.248\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.83\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e20.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e98.36\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e54\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-1.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.262\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e17.39\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e60\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-2.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.291\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e32.28\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e78\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.305\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e4.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e122.60\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e96L\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.87\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.82\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e33.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.381\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e8.64\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e372.26\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003csub\u003e\u003cb\u003e96C\u003c/b\u003e\u003c/sub\u003e\u003cb\u003e/Gr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-4.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e38.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.295\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e17.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e \u003cp\u003e347.47\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable S2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eExcited state properties of the parent C\u003csub\u003e54\u003c/sub\u003e NG.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExcited state\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWavelength\u003c/p\u003e \u003cp\u003e(nm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eOsc. Strength\u003c/p\u003e \u003cp\u003e(a.u.)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eE\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eTransitions\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003edS\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e \u003cp\u003e(a.u.)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e468.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-1\u0026rarr;LUMO (50%), HOMO\u0026rarr;L\u0026thinsp;+\u0026thinsp;1 (48%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e459.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.159\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-1\u0026rarr;L\u0026thinsp;+\u0026thinsp;1 (14%), HOMO\u0026rarr;LUMO (83%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.407\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e406.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.034\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-2\u0026rarr;LUMO (20%), HOMO\u0026rarr;L\u0026thinsp;+\u0026thinsp;1 (12%), HOMO\u0026rarr;L\u0026thinsp;+\u0026thinsp;2 (63%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.459\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e391.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.495\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-1\u0026rarr;L\u0026thinsp;+\u0026thinsp;1 (78%), HOMO\u0026rarr;LUMO (11%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.366\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e5\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e389.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.528\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-2\u0026rarr;LUMO (17%), H-1\u0026rarr;LUMO (40%), HOMO\u0026rarr;L\u0026thinsp;+\u0026thinsp;1 (35%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.771\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e6\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e376.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.139\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-2\u0026rarr;L\u0026thinsp;+\u0026thinsp;1 (23%), H-1\u0026rarr;L\u0026thinsp;+\u0026thinsp;2 (67%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.719\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e7\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e365.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.269\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-2\u0026rarr;LUMO (59%), HOMO\u0026rarr;L\u0026thinsp;+\u0026thinsp;2 (29%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3.231\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e8\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e351.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.113\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eH-2\u0026rarr;L\u0026thinsp;+\u0026thinsp;1 (65%), H-1\u0026rarr;L\u0026thinsp;+\u0026thinsp;2 (19%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.301\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e9\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e346.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eHOMO\u0026rarr;L\u0026thinsp;+\u0026thinsp;3 (93%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.154\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e342.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eHOMO\u0026rarr;L\u0026thinsp;+\u0026thinsp;4 (85%)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.014\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eEnergy transfer shows a similar size- and symmetry-related trends to charge transfer. The high-symmetry NGs C42 and C96C show high transfer rates for all three considered processes: electron, hole, and energy transfers. To disentangle the effect of symmetry on the charge and energy transfer processes, we will focus the discussion on the two prevalent processes in NG/Gr nanojunctions: PHT and FRET.\u003c/p\u003e \u003cp\u003eAll of the calculated charge and energy transfer rates are on the order of 10\u003csup\u003e13\u003c/sup\u003e-10\u003csup\u003e15\u003c/sup\u003e s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. The most prevalent CT process is the hole transfer, which occurs at the highest rate for every considered system. This is in agreement with experiment, in which the PHT prevails in NG/Gr heterostructures [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. PET is predicted to be significant only in centrosymmetric NGs, and negligible when the symmetry is broken.\u003c/p\u003e \u003cp\u003eFRET rate exceeds both CT rates for the smallest centrosymmetric C\u003csub\u003e42\u003c/sub\u003e/Gr heterojunction. For longer C\u003csub\u003e2\u003c/sub\u003e systems its rate is decreased. However, unlike PET, steadily increase with NG size, reaching the highest value for the C96\u003csub\u003eC\u003c/sub\u003e/Gr interface.\u003c/p\u003e \u003cp\u003eIn both cases of PHT and FRET, the rates are high for C\u003csub\u003e3\u003c/sub\u003e-symmetric NGs, and strongly decrease for C\u003csub\u003e2\u003c/sub\u003e compounds. In case of PHT, this symmetry-driven deterioration is overcome when the size is increased from 42 to 78 atoms, shown for C\u003csub\u003e78\u003c/sub\u003e by γ\u003csub\u003ePHT\u003c/sub\u003e(C\u003csub\u003e78\u003c/sub\u003e/Gr) \u0026gt; γ\u003csub\u003eF\u003c/sub\u003e(C\u003csub\u003e42\u003c/sub\u003e/Gr), and similarly for larger NGs. For FRET, none of the calculated C\u003csub\u003e2\u003c/sub\u003e-symmetric NGs exceed the transfer rate of C\u003csub\u003e42\u003c/sub\u003e. Therefore, the additional symmetry appears to play a bigger role in case of PHT than FRET. This tells us that if a junction with quenched energy transfer is desirable, then the use of a larger NG may be beneficial.\u003c/p\u003e"},{"header":"Summary and conclusions","content":"\u003cp\u003eWe identified several characteristics of NGs that influence the transfer rates: the HOMO-LUMO gap, excited state properties, governed to a degree by the lateral size and symmetry of the NG fragment. We were also able to discern the graphene Fermi velocity near the Dirac point as a parameter influencing the optimal energy transfer rate. With the possibility of Fermi velocity engineering by doping, strain, or applying an external potential [\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e, \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e], also the Gr part of the nanojunction can be tailored, if needed, to an application with a specific NG.\u003c/p\u003e \u003cp\u003eThe study provides a thorough analysis of the observed trends, including the impact of parameters appearing in the rate equations for all three processes.\u003c/p\u003e \u003cp\u003eThe current computational study can serve as a guideline as to the achievable properties and limits of these and similar systems. The synthesis of large NGs, with up to at least 132 atoms, over 30 \u0026Aring; lateral size, and beyond can be achieved [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], with a number of geometries going beyond the C\u003csub\u003e3\u003c/sub\u003e and elongated C\u003csub\u003e2\u003c/sub\u003e shapes analyzed here.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eConceptualization, S.O.; methodology, O. S. and S.O.; validation, formal analysis and investigation M. W. and M. L.; resources,M. W., M. L., and S.O.; data curation, S.O.; writing\u0026mdash;original draft preparation, M. W., M. L.. and S.O.; writing\u0026mdash;reviewand editing, M. W., M. L. O. S. and S.O.; supervision, S.O.; project administration, S.O.; funding acquisition, S.O. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgments\u003c/h2\u003e \u003cp\u003eWe gratefully acknowledge Polish high-performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2023/016865 and for awarding this project access to the LUMI supercomputer, owned by the EuroHPC Joint Undertaking, hosted by CSC (Finland) and the LUMI consortium through PLL/2023/05/016760. This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90254). M.L. acknowledges that this article has been produced with the financial support of the European Union under the REFRESH \u0026ndash; Research Excellence For REgion Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048 via the Operational Programme Just Transition. S. O. is grateful to the National Science Centre, Poland for funding (grant no. UMO/2020/39/I/ST4/01446 and UMO-2023/50/E/ST4/00197). This research was carried out with the support of the Interdisciplinary Center for Mathematical and Computational Modeling at the University of Warsaw (ICM UW) under grants no. G83-28 and GB80-24.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eX. Yan, B. Li, L.S. Li, Colloidal graphene quantum dots with well-defined structures, Acc Chem Res 46 (2013) 2254\u0026ndash;2262.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eC. Moreno, M. Vilas-Varela, B. Kretz, A. Garcia-Lekue, M.V. Costache, M. Paradinas, M. Panighel, G. Ceballos, S.O. Valenzuela, D. Pena, A. Mugarza, Bottom-up synthesis of multifunctional nanoporous graphene, Science 360 (2018) 199\u0026ndash;203.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJ. Liu, X. Feng, Synthetic Tailoring of Graphene Nanostructures with Zigzag-Edged Topologies: Progress and Perspectives, Angew Chem Int Ed Engl 59 (2020) 23386\u0026ndash;23401.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eD. Medina-Lopez, T. Liu, S. Osella, H. Levy-Falk, N. Rolland, C. Elias, G. Huber, P. Ticku, L. Rondin, B. Jousselme, D. Beljonne, J.S. Lauret, S. Campidelli, Interplay of structure and photophysics of individualized rod-shaped graphene quantum dots with up to 132 sp(2) carbon atoms, Nat Commun 14 (2023) 4728.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eP. Fantuzzi, A. Candini, Q. Chen, X. Yao, T. Dumslaff, N. Mishra, C. Coletti, K. M\u0026uuml;llen, A. Narita, M. Affronte, Color Sensitive Response of Graphene/Graphene Quantum Dot Phototransistors, The Journal of Physical Chemistry C 123 (2019) 26490\u0026ndash;26497.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eY. Yan, D. Zhai, Y. Liu, J. Gong, J. Chen, P. Zan, Z. Zeng, S. Li, W. Huang, P. Chen, van der Waals Heterojunction between a Bottom-Up Grown Doped Graphene Quantum Dot and Graphene for Photoelectrochemical Water Splitting, ACS Nano 14 (2020) 1185\u0026ndash;1195.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZ. Liu, H. Qiu, S. Fu, C. Wang, X. Yao, A.G. Dixon, S. Campidelli, E. Pavlica, G. Bratina, S. Zhao, L. Rondin, J.S. Lauret, A. Narita, M. Bonn, K. Mullen, A. Ciesielski, H.I. Wang, P. Samori, Solution-Processed Graphene-Nanographene van der Waals Heterostructures for Photodetectors with Efficient and Ultralong Charge Separation, J Am Chem Soc 143 (2021) 17109\u0026ndash;17116.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eX. Yu, S. Fu, M. Mandal, X. Yao, Z. Liu, W. Zheng, P. Samori, A. Narita, K. Mullen, D. Andrienko, M. Bonn, H.I. Wang, Tuning interfacial charge transfer in atomically precise nanographene-graphene heterostructures by engineering van der Waals interactions, J Chem Phys 156 (2022) 074702.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZ. Liu, S. Fu, X. Liu, A. Narita, P. Samori, M. Bonn, H.I. Wang, Small Size, Big Impact: Recent Progress in Bottom-Up Synthesized Nanographenes for Optoelectronic and Energy Applications, Adv Sci (Weinh) 9 (2022) e2106055.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS. Cao, J. Wang, F. Ma, M. Sun, Charge-transfer channel in quantum dot-graphene hybrid materials, Nanotechnology 29 (2018) 145202.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eL.Y. Hsu, W. Ding, G.C. Schatz, Plasmon-Coupled Resonance Energy Transfer, J Phys Chem Lett 8 (2017) 2357\u0026ndash;2367.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS. Osella, M. Wang, E. Menna, T. Gatti, (INVITED) Lighting-up nanocarbons through hybridization: Optoelectronic properties and perspectives, Optical Materials: X 12 (2021).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS. Qiao, M. Di, J.-X. Jiang, B.-H. Han, Conjugated porous polymers for photocatalysis: The road from catalytic mechanism, molecular structure to advanced applications, EnergyChem 4 (2022).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eM. Wang, M. Langer, R. Altieri, M. Crisci, S. Osella, T. Gatti, Two-Dimensional Layered Heterojunctions for Photoelectrocatalysis, ACS Nano 18 (2024) 9245\u0026ndash;9284.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eT. F\u0026ouml;rster, Zwischenmolekulare Energiewanderung und Fluoreszenz, Annalen der Physik 437 (1948) 55\u0026ndash;75.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eR.R. Chance, A. Prock, R. Silbey, Molecular Fluorescence and Energy Transfer Near Interfaces, in: I. Prigogine, S.A. Rice (Eds.), Advances in Chemical Physics, 1978, pp. 1\u0026ndash;65.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eB.N.J. Persson, N.D. Lang, Electron-hole-pair quenching of excited states near a metal, Physical Review B 26 (1982) 5409\u0026ndash;5415.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eŠ. Kos, M. Achermann, V.I. Klimov, D.L. Smith, Different regimes of F\u0026ouml;rster-type energy transfer between an epitaxial quantum well and a proximal monolayer of semiconductor nanocrystals, Physical Review B 71 (2005).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eP.L. Hern\u0026aacute;ndez-Mart\u0026iacute;nez, A.O. Govorov, Exciton energy transfer between nanoparticles and nanowires, Physical Review B 78 (2008).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eR.S. Swathi, K.L. Sebastian, Distance dependence of fluorescence resonance energy transfer, Journal of Chemical Sciences 121 (2009) 777\u0026ndash;787.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eA. Campion, A.R. Gallo, C.B. Harris, H.J. Robota, P.M. Whitmore, Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances, Chemical Physics Letters 73 (1980) 447\u0026ndash;450.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eA.P. Alivisatos, D.H. Waldeck, C.B. Harris, Nonclassical behavior of energy transfer from molecules to metal surfaces: Biacetyl(3nπ*)/Ag(111), The Journal of Chemical Physics 82 (1985) 541\u0026ndash;547.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJ. Hill, S.Y. Heriot, O. Worsfold, T.H. Richardson, A.M. Fox, D.D.C. Bradley, Controlled F\u0026ouml;rster energy transfer in emissive polymer Langmuir-Blodgett structures, Physical Review B 69 (2004).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eE. Malic, H. Appel, O.T. Hofmann, A. Rubio, Forster-Induced Energy Transfer in Functionalized Graphene, J Phys Chem C Nanomater Interfaces 118 (2014) 9283\u0026ndash;9289.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eE. Malic, H. Appel, O.T. Hofmann, A. Rubio, Energy-transfer in porphyrin‐ functionalized graphene, physica status solidi (b) 251 (2014) 2495\u0026ndash;2498.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eO.Y. Semchuk, O.O. Havryliuk, A.A. Biliuk, Inductive-resonance energy transfer in hybrid carbon nanostructures, Himia, Fizika ta Tehnologia Poverhni 15 (2024) 328\u0026ndash;339.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eR.A. Marcus, N. Sutin, Electron transfers in chemistry and biology, Biochimica et Biophysica Acta (BBA) - Reviews on Bioenergetics 811 (1985) 265\u0026ndash;322.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eG. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys Rev B Condens Matter 47 (1993) 558\u0026ndash;561.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eG. Kresse, J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium, Phys Rev B Condens Matter 49 (1994) 14251\u0026ndash;14269.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eG. Kresse, J. Furthm\u0026uuml;ller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational Materials Science 6 (1996) 15\u0026ndash;50.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eG. Kresse, J. Furthm\u0026uuml;ller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys Rev B Condens Matter 54 (1996) 11169\u0026ndash;11186.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJ.P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys Rev Lett 77 (1996) 3865\u0026ndash;3868.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eG. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Physical Review B 59 (1999) 1758\u0026ndash;1775.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS. Grimme, J. Antony, S. Ehrlich, H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J Chem Phys 132 (2010) 154104.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS. Grimme, S. Ehrlich, L. Goerigk, Effect of the damping function in dispersion corrected density functional theory, J Comput Chem 32 (2011) 1456\u0026ndash;1465.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eR.F.W. Bader, A Bond Path: A Universal Indicator of Bonded Interactions, The Journal of Physical Chemistry A 102 (1998) 7314\u0026ndash;7323.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eG. Henkelman, A. Arnaldsson, H. J\u0026oacute;nsson, A fast and robust algorithm for Bader decomposition of charge density, Computational Materials Science 36 (2006) 354\u0026ndash;360.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eM.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, G.A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A.V. Marenich, J. Bloino, B.G. Janesko, R. Gomperts, B. Mennucci, H.P. Hratchian, J.V. Ortiz, A.F. Izmaylov, J.L. Sonnenberg, Williams, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V.G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J.A. Montgomery Jr., J.E. Peralta, F. Ogliaro, M.J. Bearpark, J.J. Heyd, E.N. Brothers, K.N. Kudin, V.N. Staroverov, T.A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A.P. Rendell, J.C. Burant, S.S. Iyengar, J. Tomasi, M. Cossi, J.M. Millam, M. Klene, C. Adamo, R. Cammi, J.W. Ochterski, R.L. Martin, K. Morokuma, O. Farkas, J.B. Foresman, D.J. Fox, Gaussian 16 Rev. C.01, Wallingford, CT, 2016.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eA.V. Krukau, O.A. Vydrov, A.F. Izmaylov, G.E. Scuseria, Influence of the exchange screening parameter on the performance of screened hybrid functionals, J Chem Phys 125 (2006) 224106.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eB. Baumeier, J. Kirkpatrick, D. Andrienko, Density-functional based determination of intermolecular charge transfer properties for large-scale morphologies, Phys Chem Chem Phys 12 (2010) 11103\u0026ndash;11113.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eH. Oberhofer, K. Reuter, J. Blumberger, Charge Transport in Molecular Materials: An Assessment of Computational Methods, Chem Rev 117 (2017) 10319\u0026ndash;10357.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eC. Musumeci, S. Osella, L. Ferlauto, D. Niedzialek, L. Grisanti, S. Bonacchi, A. Jouaiti, S. Milita, A. Ciesielski, D. Beljonne, M.W. Hosseini, P. Samori, Influence of the supramolecular order on the electrical properties of 1D coordination polymers based materials, Nanoscale 8 (2016) 2386\u0026ndash;2394.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eM. Dobbelin, A. Ciesielski, S. Haar, S. Osella, M. Bruna, A. Minoia, L. Grisanti, T. Mosciatti, F. Richard, E.A. Prasetyanto, L. De Cola, V. Palermo, R. Mazzaro, V. Morandi, R. Lazzaroni, A.C. Ferrari, D. Beljonne, P. Samori, Light-enhanced liquid-phase exfoliation and current photoswitching in graphene-azobenzene composites, Nat Commun 7 (2016) 11090.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS. Osella, S. Knippenberg, Environmental effects on the charge transfer properties of Graphene quantum dot based interfaces, International Journal of Quantum Chemistry 119 (2018).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS.F. Nelsen, S.C. Blackstock, Y. Kim, Estimation of inner shell Marcus terms for amino nitrogen compounds by molecular orbital calculations, Journal of the American Chemical Society 109 (1987) 677\u0026ndash;682.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJ. Blumberger, Recent Advances in the Theory and Molecular Simulation of Biological Electron Transfer Reactions, Chem Rev 115 (2015) 11191\u0026ndash;11238.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eM. Kaźmierczak, S. Giannini, S. Osella, Photoinduced energy and electron transfer at graphene quantum dot/azobenzene interfaces, Journal of Materials Chemistry C 12 (2024) 143\u0026ndash;153.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eR.L. Martin, Natural transition orbitals, The Journal of Chemical Physics 118 (2003) 4775\u0026ndash;4777.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJ.R.F. Lima, Controlling the energy gap of graphene by Fermi velocity engineering, Physics Letters A 379 (2015) 179\u0026ndash;182.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eA. Diaz-Fernandez, L. Chico, J.W. Gonzalez, F. Dominguez-Adame, Tuning the Fermi velocity in Dirac materials with an electric field, Sci Rep 7 (2017) 8058.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"theoretical-chemistry-accounts","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"tcac","sideBox":"Learn more about [Theoretical Chemistry Accounts](http://link.springer.com/journal/214)","snPcode":"214","submissionUrl":"https://submission.nature.com/new-submission/214/3","title":"Theoretical Chemistry Accounts","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Charge transfer, Energy transfer, Heterojunction, Nanographene, Computational approach","lastPublishedDoi":"10.21203/rs.3.rs-5199549/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5199549/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe charge and energy transfer processes in photoexcited 0D/2D donor/graphene heterojunctions occur through multiple different pathways. A donor deexcitation event occurring in the most prevalent F\u0026ouml;rster energy transfer mechanism (strongly favored over Dexter transfer in van der Waals heterojunctions) forbids the charge transfer from occurring, thus creating a competition between the two processes. By applying a robust computational approach, we describe the two processes from first principles, and quantify their rates using F\u0026ouml;rster and Marcus theories. We consider nanojunctions where the donor are nanographenes with varying size and symmetry, and discern important trends, e.g. the symmetry-induced quenching, or the enhancement due to increased size. We find that heterojunctions where nanographenes do not have a center of symmetry show decreased photoinduced hole and energy transfer rates, which can then be recovered by increasing the delocalization length, whereas for centrosymmetric nanographenes both hole and energy transfer processes are enhanced. However, the hole transfer rate dominates over the energy transfer process, providing a new computation-driven design principle for obtaining a high-charge transfer junction with minimized contribution of the competing energy transfer.\u003c/p\u003e","manuscriptTitle":"Unraveling the competition between charge and energy transfer in 0D/2D nanographene-graphene heterojunctions","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-11-25 06:11:20","doi":"10.21203/rs.3.rs-5199549/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-11-05T15:09:14+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-11-05T11:49:51+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-10-21T07:50:08+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"327532328449888017973407106638179511080","date":"2024-10-08T12:37:46+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"116192514533649151182489347905779867529","date":"2024-10-06T13:57:57+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-10-06T09:51:32+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-10-04T01:56:30+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-10-04T01:55:21+00:00","index":"","fulltext":""},{"type":"submitted","content":"Theoretical Chemistry Accounts","date":"2024-10-03T16:13:48+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"theoretical-chemistry-accounts","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"tcac","sideBox":"Learn more about [Theoretical Chemistry Accounts](http://link.springer.com/journal/214)","snPcode":"214","submissionUrl":"https://submission.nature.com/new-submission/214/3","title":"Theoretical Chemistry Accounts","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"c0c917c6-9778-4b7e-aadb-7524442ab35d","owner":[],"postedDate":"November 25th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-12-30T16:07:08+00:00","versionOfRecord":{"articleIdentity":"rs-5199549","link":"https://doi.org/10.1007/s00214-024-03166-1","journal":{"identity":"theoretical-chemistry-accounts","isVorOnly":false,"title":"Theoretical Chemistry Accounts"},"publishedOn":"2024-12-27 15:57:23","publishedOnDateReadable":"December 27th, 2024"},"versionCreatedAt":"2024-11-25 06:11:20","video":"","vorDoi":"10.1007/s00214-024-03166-1","vorDoiUrl":"https://doi.org/10.1007/s00214-024-03166-1","workflowStages":[]},"version":"v1","identity":"rs-5199549","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5199549","identity":"rs-5199549","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}