The Role of Orbital Symmetries in Enforcing Ferromagnetic Ground State in Mixed Radical Dimers

One of the first steps in designing ferromagnetic (FM) molecular materials of p-block radicals is the suppression of covalent radical–radical interactions that stabilize a diamagnetic ground state. In this contribution, we demonstrate that FM coupling between p-block radicals can be achieved by constructing mixed dimers from different radicals with differing symmetries of their singly occupied molecular orbitals. The applicability of this approach is demonstrated by studying magnetic interactions in organic radical dimers built from different derivatives of the well-known phenalenyl radical. The calculated enthalpies of dimerization for different homo- and heterodimers show that the formation of a mixed dimer with FM coupling is favored compared to the formation of homodimers with antiferromagenetic (AFM) coupling. We argue that cocrystallization of radicals with specifically tuned morphologies of their singly occupied molecular orbitals is a feasible and promising approach in designing new organic magnetic materials.

projection M S = 0 were optimized. The corresponding exchange coupling constant was then evaluated using the Yamaguchi projection S33-S36 where E S and E T are the energies of the singlet and triplet state, respectively, E LS and E HS are the energies of the LS and HS states, respectively, and S 2 LS and S 2 HS are the expectation values of theŜ 2 operator evaluated on the Kohn-Sham determinant. E LS and E HS in equation (1) are evaluated at a fixed geometry, but the equation can also be generalized into an adiabatic situation where the geometries are allowed to relax S37 assuming that i) the energy and geometry evaluated for the HS state are reasonable approximations of the triplet energy and geometry, and ii) the geometry optimized for the LS state is a reasonable approximation of the singlet geometry. The first assumption certainly seems valid as the HS state is single-determinantal and therefore should pose no problem to DFT; the latter assumption inevitably introduces some approximation. As J is equal to the singletriplet splitting, the energy of the singlet state is given as where the subscripts indicate the state and the superscripts indicate the geometry the energy or S 2 value is evaluated in. Using this result, the exchange coupling constants were evaluated in the LS geometry (J LS ), in the HS geometry (J HS ), and using the adiabatic singlet-triplet splitting (J adiabatic ) as

Dimerization enthalpies
The dimerization energies (∆E) and enthalpies (∆H) were evaluated using two different approaches; in validation studies (vide infra) all terms in the expressions were evaluated at DFT level, and in the final calculations energy terms of the HS states were evaluated at DLPNO-CCSD(T) level using the DFT optimized geometries, and the exchange coupling constants and enthalpy corrections to the energies were evaluated at the DFT level. In both cases the basis set superposition error (E BSSE ) was estimated using the counterpoise method S38 either at the DFT or DLPNO-CCSD(T) level depending on which energies were used in the expressions. Using only DFT energies, the dimerization energies and enthalpies in the LS and HS states are given as It should be noted that ∆E HS is calculated purely from DLPNO-CCSD(T) energies (although using DFT optimized geometries), whereas ∆E LS is evaluated using the exchange coupling constants calculated at the BS DFT level.

Validation studies
In order to produce reliable results in calculations on the hypothetical 2-3 dimer, several functionals were first validated against experimental data. Based on the results presented below, the range-separated LC-ωPBE hybrid functional with the DFT-D3 dispersion correction and BJ damping function was chosen for the final calculations.

Exchange coupling constants
The exchange coupling constants calculated using various functionals were compared to the experimentally measured values. The calculated values of the exchange coupling constants depend both on the geometry used for the calculations and the ability of the chosen XC functional to correctly estimate the energy differences between different spin states within the context of BS formalism. It is well known that values of exchange coupling constants have strong dependence on the XC functional, S37,S39-S48 whereas the geometry optimizations tend to compensate for the functional error. S49 In order to eliminate the degrees of freedom related to geometry optimizations, the calculations were carried out as single point energy evaluations on geometries extracted from experimental crystal structures. Based on SQUID measurements, S50-S52 the exchange coupling constant in 2-2 has been determined as roughly −1390 cm −1 . The measurement has, however, been carried out in the presence of paramagnetic impurities and the experimental reference should therefore be considered only as a rough estimate. The exchange coupling constant in 2-2 has also been determined from peak intensities of the EPR spectrum, giving a value of −2910 cm −1 , which is more than twice the magnitude of the SQUID value. S53 Due to the uncertainties related to the experimental value of the exchange coupling constant in 2-2, in addition to the 2-2 dimer, validation calculations were also carried out on the dimer of 3,5-di-tert-butyl-8-para-bromophenyl-6oxophenalenoxyl (4), which is structurally similar to 3-3 and for which an exchange coupling constant of −267 cm −1 has been reliably determined from SQUID data. S54  The calculated exchange coupling constants are listed in Table S1. It is immediately clear that all values calculated for 2-2 are considerably larger in magnitude than the value estimated from SQUID measurements. The value calculated with the range-separated LC-ωPBE functional is closest to the SQUID value but its magnitude is still more than twice that of the experimental value. The LC-ωPBE value is, however, relatively close to the value extracted from EPR measurements. All functionals do correctly produce the qualitative aspects of the interaction: the strong AFM exchange. In the case of the 4-4 dimer, all calculated values are much closer to experiment. The best match is again obtained with LC-ωPBE, and the value calculated with the global hybrid PBE0 is also extremely close to experiment. However, all tested functionals provide a reasonable estimate of the exchange coupling constant in this case.

Geometries and dimerization enthalpies
The structures of the monomer 2 and the 2-2 dimer in its LS state were optimized using different XC functionals. The C-C distance between the central carbons of the two monomers in the dimer geometry were compared with the value in the experimental crystal structure (3.201 Å). S52 This was chosen as the most important structural parameter as the radicalradical exchange interaction is usually extremely sensitive to the distance between the radicals. The calculated dimerization enthalpies were also compared to those measured experimentally in solution. The enthalpy has been determined both from UV-Vis measurements (−31.4 kJ mol −1 , −36.8 kJ mol −1 ) S55,S56 and from the EPR spectrum (−39.9 kJ mol −1 ). S56 The calculated values are listed in Table S2. Only the M06 and M06-2X functionals are able to produce reasonable estimates of the central C-C distance without the inclusion of some dispersion correction. In the case of all other functionals, the dimer dissociates if dispersion is not corrected for. The best estimate of the C-C distance is provided with the B3LYP-D3 functional, and the functionals LC-ωPBE-D3, LC-ωPBE-D3BJ, M06-2X, M06-2X-D3, PBE0-D3, and PBE0-D3BJ all give an error of less than 0.01 Å.
The calculated dimerization enthalpies are either overbinding, or repulsive in the case of dimers which dissociate. The closest estimate to the experimental values is obtained with CAM-B3LYP-D3 (with an overestimated C-C distance) and the M06-2X functional. In both cases, the magnitude of the dimerization enthalpy is still nearly double the experimental values. The large deviations could results from solvent interactions not accounted for in the calculations. Preliminary calculations were carried out using a polarizable continuum model to account for electrostatic solvent interactions, but no significant differences in the results was observed. This does not, however, rule out the possibility of stabilizing non-electrostatic solvent interactions. Based on these observations, the ability of the chosen XC functional to correctly predict the geometry was made the main criteria, and in the final calculations the dimerization enthalpy was calculated using DLPNO-CCSD(T) energies evaluated on the DFT optimized geometries. This approach should considerably reduce the dependence of It is worth noting here that Kertesz and co-workers have optimized the geometry of 2-2 using the M05-2X functional without any dispersion correction and obtained a dimerization energy (including zero point and counterpoise corrections) of −32.9 kJ mol −1 and a central C-C distance of 3.209 Å, both in very good agreement with experiments. S57 In the present work, the dimerization energy calculated using the M06-2X functional (which is an improved version of M05-2X) without any dispersion correction is −83.0 kJ mol −1 (the respective enthalpy is −74.0 kJ mol −1 ). Thus, the value presented here is considerably larger than that calculated by Kertesz and co-workers. We own these differences to the use of spin-projection in our calculations and the larger basis sets. In the present work, initial calculations were carried out using the smaller def2-SVP S58 and TZVP S59 basis sets but after observing con-S9 siderable deviations between the results, all DFT data were re-calculated using the larger def2-TZVP basis. S10 Additional computational data