Exchange controlled triplet fusion in metal–organic frameworks

Triplet-fusion-based photon upconversion holds promise for a wide range of applications, from photovoltaics to bioimaging. The efficiency of triplet fusion, however, is fundamentally limited in conventional molecular and polymeric systems by its spin dependence. Here, we show that the inherent tailorability of metal–organic frameworks (MOFs), combined with their highly porous but ordered structure, minimizes intertriplet exchange coupling and engineers effective spin mixing between singlet and quintet triplet–triplet pair states. We demonstrate singlet–quintet coupling in a pyrene-based MOF, NU-1000. An anomalous magnetic field effect is observed from NU-1000 corresponding to an induced resonance between singlet and quintet states that yields an increased fusion rate at room temperature under a relatively low applied magnetic field of 0.14 T. Our results suggest that MOFs offer particular promise for engineering the spin dynamics of multiexcitonic processes and improving their upconversion performance.


Supplementary Note 1. The MFE calculation of triplet fusion
The fusion process can be kinetically described as where the fusion rate (ks) is governed by a spin character of the triplet pair state (TT). The plots in Fig. 2 are calculated by the simple Merrifield model 1 to show the effect of the intertriplet exchange coupling. The parameters used in the calculations are k-1/ks= 1, D= 8.5 µeV, and E= -1.1 µeV, based on a combination of fits to EPR and MFE data. The simple Merrifield model assumes stationary states of the spin Hamiltonian, only valid for a pair with long lifetime. We used the Johnson-Merrifield model that employs the full density matrix of the triplet pair spin states to better fit the experimental results 2 . The time dependence of the density matrix is, The first term is the time evolution of the density matrix by spin Hamiltonian. The second is the dissociation of a triplet pair, the third is the fusion from a triplet pair to a singlet exciton, and the last term is the triplet pair formation. The density matrix was solved numerically with the steady-state condition ( )* )+ = 0) to calculate the fusion rate. The full elements of the spin Hamiltonian can be found in Ref. 3 . We have generated 1000 random molecular orientations and averaged the results to account for the polycrystalline sample.
To fit the data in Fig. 4a, we combined two MFE curves (with/without the exchange interaction) at a 16: 84 ratio. For non-exchange coupled MFE, kS= 1×10 8 s -1 , k-1= 1.7×10 8 s -1 , J= 0 , D= 8.5 µeV and E= -1.17 µeV were used. For exchange-coupled MFE, kS= 5×10 9 s -1 , k-1= 3.3×10 9 s -1 , J= -6.1 µeV, D= 8.5 µeV and E= -1.17 µeV. Faster kS and k-1 were used for the exchange-coupled pair, because an excitonic hopping changes the exchange interaction that reduces the TT pair lifetime under a specific exchange coupling. In contrast, many sufficiently separated triplet pairs have negligible J, so the hopping in nonexchange coupled regime does not affect the TT pair lifetime. Repeated experiments at different locations on a sample, and samples prepared under different conditions yielded MFE curves with the same peak positions, but the overall signal can shift as shown in Fig. S6. The peak positions are determined by J, D, and E. The various rates described above affect the intensities of the peaks. We propose that one possible explanation for the varying offset in the experimental data is different degrees of triplet-charge annihilation, and a monotonic Lorentzian curve 4 was added to simulate triplet-charge annihilation and correct the shift up to 2.5%.
The thickness of each MOF sample was observed to be highly spatially non-uniform, contributing to variation in the measured optical absorption, PLQY, and EQE.
Defining the maximum efficiency as 100%, the measurement of the internal quantum yield of upconversion, =6 > , as described in Methods, yielded (3.72´10 -2 )% and (1. The ;< 's of NU-1000 and NU-901 are measured (see Methods) to be (2.06±0.12)% and (0.26±0.09)%, respectively, with the error determined from measurements at three different locations on the sample. The range of measured =6,/ > for NU-1000 and NU-901 was (0.54% -1.8%) and (0.14%-0.45%), respectively. Noting that the efficiency scales inversely with absorption, we assumed that the peak measured =6,/ > was the most accurate quantification of the internal processes given the presence of reabsorption losses.

Supplementary Note 3. Monte Carlo simulation of the average time spent at various separation distances
The considered kinetic processes can be illustrated as, where kh, kS, kdec are hopping rate, fusion rate, and spin relaxation rate. Note that it is depicted as 1D lattice for simplicity, but the calculations were done with 3D lattice parameters of NU-1000, NU-901, and anthracene crystals. Triplet pairs are allowed to fuse between the nearest-neighbors and they are set to dissociate beyond a specific separation. The initial condition was 3.8 nm separation, and it has a marginal effect on the results. Triplet pairs are assumed to fuse at the nearest-neighbor (NN) configuration and dissociate beyond 4 nm separation. We repeated the process until we collect 1000 trajectories of fusion, and averaged the time spent at each separation distance. The used parameters are summarized in Table 3, and the results are shown in Fig. S4. Table 3. Model parameters used in the triplet-pair trajectory simulations. The NN and 2nd NN hopping distances of NU-1000 are 1 nm and 1.7 nm, respectively. The 2nd NN hopping includes both ab-plane and c-axis directions. Anthracene hopping rates are adopted from a reference 27 and the decoherence rate is based on the decay rates of quantum beating in tetracene 28 . For the hopping rates of MOF, we use the calculated values with PBE0 functional.

Supplementary Note 4. DFT calculation of hopping rates and exchange couplings
The hopping rates were calculated using Fermi's golden rule where ?@A is the coupling that mediates the triplet hopping and BC is the Franck-Condon weighted density of states (FCWD). In the hopping integral, we include up to the second order, charge transfer (CT) terms, as they have been shown to be important 6,7 ?
The FCWD was approximated using Marcus theory 8,9 where Δ is the energy gap and F is the reorganization energy. Since the donor and the acceptor are symmetric, we have Δ = 0. Assuming that F is dominated by intramolecular reorganization, we can write it in terms of the monomer properties where I and I are the energy and the geometry of the state , respectively.
The electronic structure calculations were performed using Q-Chem 5.1 software package. 10 We started with the structure of the NU-1000 MOF in the MOF Lab web module of Maxime Usdin. 11 In order to minimize the computational cost, we cropped just one or two organic linkers that were the most relevant to the problem: the monomer, the nearest neighbor dimer, and the second nearest neighbor dimer. The change imbalance was repaired by capping the carboxyl groups with hydrogen atoms. Then, a constrained geometry optimization was performed with the carboxyl oxygen atoms as the fixed atoms. The constraints ensured that the organic linkers did not move with respect to each other.
The hopping integral was calculated using constrained density functional theory (CDFT). 12 We used the ground state dimer geometries optimized using the B3LYP exchange-correlation functional 13,14 and the 6-31G* basis set. [15][16][17][18] A total of four diabatic states were calculated: Symmetric orthogonalization was performed in a pairwise manner to avoid the mixing of the excitonic states and the CT states. For example, we orthogonalized just the 2×2 matrix in the basis of {| * ⟩, | # A E ⟩} in calculating the coupling ⟨ # E | | * ⟩. The CDFT calculations were repeated on the same geometry with a range of functionals: CAM-B3LYP, 19 PBE, 20 PBE0, 21 LRC-ωPBEh, 22 M06-L, 23 and M06-2X. 24 To calculate the reorganization energy, the monomer geometries were optimized in the S0 and the T1 states, and the S0 and the T1 energies were calculated at each geometry. We obtained the S0 and the T1 states using restricted and unrestricted density functional theory (RDFT and UDFT), respectively. The excited state optimization and reorganization energy calculation were repeated with each of the abovementioned functionals.
The simulated reorganization energies, hopping integrals, and hopping rates are summarized in Table 1.
Not surprisingly, the hopping rates exhibit a strong dependence on the fraction of exact exchange. Whereas PBE predicts the nearest neighbor hopping rate of 4.0×10 10 s −1 , PBE0 predicts the rate to be almost two orders of magnitude smaller at 6.6×10 9 s −1 . The decrease in the hopping rate is a combined effect of a decrease in the couplings and an increase in the reorganization energy. In addition, it is concerning that LRC-ωPBEh predicts the second nearest neighbor hopping integral to be zero. To be precise, the value is smaller than the convergence tolerance of the electronic structure calculations (10 −11 au).
We suspect that the application of range-separated hybrids, such as CAM-B3LYP and LRC-ωPBEh, in conjunction with CDFT might be problematic. It has been shown that global hybrids, such as B3LYP and PBE0, can give more accurate excitation energies than range-separated hybrids when used in conjunction with ΔSCF. 25,26 In particular, range-separated hybrids tend to overestimate the energies. The reason might be that range-separated hybrids overcompensate for the delocalization error which orbital relaxation corrects to some extent. A similar situation arises in the case of CDFT, which is also a state-specific method. Then, we expect that range-separated hybrids would underestimate the couplings and overestimate the energies. Furthermore, B3LYP and PBE0 can be expected to give the most reliable estimates of the energetics. The triplet-triplet exchange was estimated by the coupling where m1 and m2 are the spins of the two monomers. There is not a simple equivalence between the spin-1 particle wavefunction in the Heisenberg model and the spin-1/2 particle wavefunction in electronic structure theory. However, we expect that the above coupling would estimate the triplet-triplet exchange to the order of magnitude. The coupling itself was calculated using the same procedure as the hopping integrals.
The simulated couplings are summarized in Table 2. Like the case of the hopping integrals, CAM-B3LYP and LRC-ωPBEh predict the couplings to be zero within the tolerance. Again, we suspect that the application of range-separated hybrids in conjunction with CDFT might be leading to a gross underestimation of the coupling.

Supplementary Note 5. Optimization of MOF design
As noted in the main text, the magnitude of the singlet-quintet resonance observed in NU-1000 is 2%.
To realize the potential of singlet-quintet mixing, we expect that the tailorability of MOFs can be exploited to engineer larger resonances. Figure S5 predicts trends in the magnitude of the singlet-quintet resonance as a function of three key parameters: the rate of TT pairs separating back into individual triplet excitons, k-1; the formation rate of singlet excitons from TT pairs, kS; and the zero-field splitting parameter, D, which measures the coupling of the singlet and quintet states. Three trends are observed.
(i) The resonance increases when we retard the separation of TT states back into individual triplets ( Fig. S5(a)) (ii) The resonance increases when we increase the rate of singlet exciton formation from TT states ( Fig. S5(b)) (iii) Increasing the coupling of singlet and quintet states increases the rate of spin mixing, which in turn can increase the magnitude of the resonance in the regime where D < J, where J is the exchange splitting (Fig. S5(c)).
The first two trends have a significant impact on the resonance and emphasize the potential importance of engineering the excitonic energy levels and employing exothermic fusion materials. When optimized, the peak resonance is observed in Fig. S5(d) to increase from 2% to 65%.
There are also potential improvements that may be obtained through structural design of the MOFs. Increasing the nearest neighbor distance by engineering reticular derivatives of NU-1000 is the most obvious target for future improvements. Further separating the pyrene cores by employing larger linker arms, for instance, is likely to render the exchange interaction negligible, such that spin mixing could occur even in the absence of a magnetic field. Too much separation, however, might slow down the fusion and hopping rate. The fusion rate has a quadratic dependence on triplet density, causing separation to impare the upconversion performance at low excitation intensity. Thus, the tradeoff between triplet diffusion and singlet-quintet resonant efficiency enhancements suggests optimization of localization may depend on the excitation intensity during operation.   . Electron paramagnetic resonance (EPR) signal of NU-1000 and simulated curve with g=2.0. One characteristic feature for the EPR spectrum of a triplet state is the partially forbidden |Δms| = 2 transition that appears at g ~ 4.0 when the magnetic field orientation deviates from the principal axes of the molecule (known as a half-field transition) 29 . Such a signal is expected to be particularly pronounced for randomly oriented samples 8 . The apparent peak near 161.7 mT was assigned to be the half-field transition of the triplet exciton in the MOF. The peak position agrees with the simulation using |D| = 8.50 μeV and |E| = 1.17 μeV, further supporting the analysis we obtained from the MFE measurements. The g = 2 transitions were not observed probably because they were too broad to be separated from the baseline. The source of the broadening could be a distribution in the D and E values.     The molar volume of NU-1000 is 2220 cm 3 , and anthracene is 143 cm 3 . A TIPS-tetracene crystal 31 , one of the bulkiest singlet fission materials, has a molar volume of 546 cm 3 that is still four times lower than NU-1000. The average atomic ratio of Pt/Zr is 1.5 ± 0.6 and 4.5 ± 0.7 for NU-1000:PtOEP and NU-901:PtOEP, respectively. Please note that the PtOEP washing step after the drop casting cannot ensure the complete removal of excess PtOEP, resulting in a higher measured Pt/Zr atomic ratio by EDS than the true value for the MOF:PtOEP systems. Further, the residual PtOEP also led to inhomogeneity of the samples, as reflected by varying Pt/Zr atomic ratios in different areas.