Lanthanide-radical magnetic coupling in [LnPc$_2$]$^0$: Competing exchange mechanisms captured via ab initio multi-reference calculations

We present a computational investigation of the intramolecular exchange coupling in [LnPc$_2$]$^0$ (Ln = Tb, Dy, Ho, and Er) between the Ln$^{3+}$ 4f electrons and the spin-1/2 radical on the phthalocyanine ligands. A series of ab initio multi-configurational/multi-reference Complete/Restricted Active Space Self-Consistent-Field calculations (CASSCF/RASSCF), including non-perturbative spin--orbit coupling, were performed on [LnPc$_2$]$^0$ and on the smaller model compound [LnPz$_2$]$^0$. We find that the exchange coupling mechanisms are restricted by symmetry, but also dependent on the spin polarization effect triggered by the Pc$_2$ ligands $\pi$--$\pi^*$ excitations. The calculated exchange splittings are small, amounting to at most a few cm$^{-1}$, in disagreement with previous literature reports of strong antiferromagnetic coupling, but in good agreement with recent EPR experiments on [TbPc$_2$]$^0$. Furthermore, the coupling strength is found to decrease from [TbPc$_2$]$^0$ to [ErPc$_2$]$^0$, with decreasing number of unpaired electron spins in the lanthanide ground (Hund's rule) Russell--Saunders term.

Introduction and their conclusion that the lanthanide is strongly and antiferromagnetically coupled to the Pc 2 radical has been repeated unchallenged in review articles. 7,21,22 Recently, evidence to the contrary was derived from a single-crystal EPR experiment on [TbPc 2 ] 0 . 23 The field and angle dependent resonance frequencies were found to be consistent with a small ferromagnetic interaction described by the Ising Hamiltonian Here,S Ln denotes an effective spin of 1/2 representing the ground state doublet on Tb, and S Pc 2 denotes the real spin of the Pc 2 radical. The exchange splitting derived from the EPR measurement is J eff = 0.88 cm −1 . Note that the choice of writing the exchange coupling Hamiltonian between two pseudo-spin 1/2 as in Eq. (1) implies that 0.88 cm −1 corresponds to the energy gap between the ground ferromagnetic exchange Kramers doublet, and the first excited antiferromagnetic exchange Kramers doublet. It is clear that this small interaction is incompatible with the susceptibility data of Trojan et al. 19,20 We could find only one other published susceptibility measurement on these systems, namely for [DyPc 2 ] 0 . 24 The χT data reported by Branzoli et al. 24  calculations, suggested a ferromagnetic interaction between Pc 2 radical and Ln(III) ion. 25 However they did not report on the magnitude of the interaction. DFT   We found that the orbitals of e 1 and e 3 symmetry had a tendency to rotate out of the active space. To prevent this from happening, the 8 orbitals of e 1 and e 3 symmetry were put into an artificial symmetry class so as to disable orbital mixing with orbitals outside this class (using the "supersymmetry" keyword of MOLCAS). The validity of this approach relies on the quality of the starting orbitals. These were obtained from a state-averaged RASSCF calculation on the twofold degenerate ferromagnetic (S = 7/2) ground state of [TbPz 2 ] 0 .
This calculation did not experience the unwanted rotations and provided correct orbitals.
Before spin-orbit coupling (SOC) is considered, the exchange coupling between the lanthanide and the spin-1/2 radical can be evaluated as the energy difference between the high-spin and low-spin states that are obtained by coupling the total spin of the Hund term of the Ln 3+ ion ( 7 F for Tb 3+ , 6 H for Dy 3+ , 5 I for Ho 3+ , and 4 I for Er 3+ ) with the spin-1/2 of the radical. In each case, the state-averaging was performed over all states formally arising from the Hund term. As an example, for the [DyPc 2 ] 0 molecule, we optimize respectively S = 3 high-spin and S = 2 low-spin, with the state average including all 11 spatial components of the L = 5 Hund term 6 H of the Dy 3+ ion. We then evaluate the exchange gap as the difference between the lowest S = 3 and S = 2 energies.
Finally, SOC is introduced by matrix diagonalization in the basis of all the optimized S = 2 and S = 3 CASSCF/RASSCF wavefunctions.
We note that a similar approach was used in a recent computational study of the exchange interaction in the dimer Ce 2 (COT) 3 . 30

Results and discussion
The calculated CASSCF active natural orbitals of [LnPc 2 ] 0 are shown in the top of Fig. 2.
The Ln 4f orbitals are quasi atomic, while the spin-1/2 radical (π-SOMO) is mainly localized and evenly distributed on the C 1 atoms with nodes on the N atoms. The exchange gaps obtained from the CASSCF calculations (without SOC) are listed in Table 1  K f π represents a potential exchange integral between the π-SOMO and a 4f orbital, and J represents the total exchange strength.  Fig. 2), while the seven lanthanide 4f orbitals transform as b 2 + e 1 + e 2 + e 3 . Thus, the magnetic orbital containing the Pc 2 radical is orthogonal by symmetry to each of the magnetic orbitals of the lanthanide ion. Kinetic exchange between the magnetic orbitals is therefore forbidden and only the ferromagnetic potential exchange interaction is allowed. 31,32 In the ground Russell-Saunders term of the Ln 3+ ions considered here, the number of un-paired 4f electrons decreases with increasing overall number of 4f electrons (see Fig. 2). If we consider that each unpaired 4f electron contributes additively to the overall exchange, the latter is expected to decrease in magnitude in going from [TbPc 2 ] 0 to [ErPc 2 ] 0 , as observed in Table 1. We note in this respect a recent experimental work in which the magnetic coupling between TbPc 2 and a Ni(111) surface was also found to decrease along the series Tb-Ni > Dy-Ni > Er-Ni. 17  Tables 2 and 4). We see that the principal g-factors of the ground doublet are exactly 4 units higher than those of the next doublet. This corresponds to a spin flip of the radical electron (whose g-factor is 2), from ferromagnetic alignment in the ground doublet to antiferromagnetic alignment in the next doublet. Hence the coupling can be described as ferromagnetic. This straightforward interpretation cannot be applied to the case of Er however, because the exchange is not of Ising type there.
We now consider the effect of introducing π-π * correlation using the RASSCF method.
The calculated values of the exchange gaps before SOC are given in Table 3. These cal- The results in Table 3 show that the RASSCF exchange gaps are still ferromagnetic but smaller than the corresponding CASSCF gaps. This we interpret as the result of a competition between a new antiferromagnetic exchange pathway, opened up by activation of π-π * correlation, and the direct ferromagnetic exchange pathway already present in the CASSCF calculations.
The spin-orbit coupled RASSCF spectrum is given in Table 4. The exchange splittings are smaller than the corresponding CASSCF values ( Table 2 and Table S2) in line with the reduction of the SOC-free exchange splittings. We note in particular the value for Tb, which decreases from 6.18 cm −1 to 1.92 cm −1 , closer to the experimental value of 0.88 cm −1 . 23 Note that the recent CASSCF calculations by Pederson et al. found J eff = 8.2 cm −1 and 6.6 cm −1 for two geometries of [TbPc 2 ] 0 , which is basically the same result we obtain with our CASSCF calculation, using an active space where 4f orbitals and the Pc 2 SOMO only are considered. This seems to suggest that the π-π * excitations introduce by Pederson et al.
in their active space were not sufficient to describe the spin-polarization antiferromagnetic We attribute this breakdown of the usual model to the effect of spin polarization in the π system of the Pc 2 /Pz 2 radical. Spin polarization in radicals of conjugated π systems is a well known effect, and was invoked by McConnell to explain ferromagnetic coupling between stacked organic radicals ("McConnell's first model"). 42 Later, Yoshizawa and Hoffmann argued that these magnetic couplings can be equally well explained on the basis of interaction between the SOMO's of the organic radicals, 41 the condition for ferromagnetic coupling being again the (near) vanishing of orbital overlap.
Let us now consider the spin density distribution in the Pc 2 /Pz 2 radical. The SOMO (pictured in Fig. 2) has amplitudes on the C atoms but nodes on all the N atoms. The spin density, in the simple molecular orbital picture, is therefore positive on the carbons but zero on the nitrogens. When we allow for electron correlation in the π system (as in our RASSCF calculations), small but negative spin densities appear on the N atoms. This is illustrated numerically with Mulliken spin populations in Table 5.
An elaborate analysis of the interplay between spin polarization and exchange in [LnPc 2 ] 0 Table 5: RASSCF Mulliken spin populations ρ on N and C atoms (Fig.1). The spin-1/2 radical is mainly localized on C 1 . The small negative spin populations on the N atoms are due to the spin-polarization effect. will not be attempted here. Instead, a simple argument in the spirit of McConnell's first model will be given. Let us assume then, that the total exchange splitting can be estimated as the sum of contributions from each atom of the ligand, and that only those atoms whose spin populations are non-zero can contribute. We can also assume that atoms further away from the central lanthanide ion will have a smaller exchange interaction with it than atoms closer by. Referring to Fig. 1, the atoms closest to Ln 3+ are the 8 N 1 atoms at 2.41 Å and the 16 C 1 atoms at 3.36 Å.
In the absence of spin polarization (the CASSCF case) there is only spin density on C 1 . Since all C 1 atoms are symmetry related, the contribution from each of them to the exchange interaction must be the same. And since the overall interaction is ferromagnetic, the contribution from each C 1 atom must be ferromagnetic as well. On the other hand, when spin polarization is allowed (the RASSCF case), the N 1 atoms carry negative spin density, which will also interact with the lanthanide spin. If we may assume that this interaction is ferromagnetic, just like that of the C 1 atoms, a competition arises: On the one hand, the majority spin on C 1 atoms tries to align itself parallel to the Ln 3+ spin, favoring overall ferromagnetic coupling. On the other hand, the polarized minority spin density on N 1 atoms, with an opposite sign of spin compared with C 1 atoms, also tries to be parallel to the metal spin, favoring overall antiferromagnetic coupling. As a result, the total exchange interaction is a sum of a positive contribution from C 1 and a negative contribution from N 1 . Apart from the sign, it is not possible to determine a priori how large the contribution from N 1 is compared to that from C 1 . This can be seen from considering the two parameters that will determine the size of the contribution: the spin density on the atom and the distance from the atom to the lanthanide ion. The spin density on N 1 is smaller than on C 1 , but N 1 is closer to the lanthanide than C 1 (2.41 Å vs. 3.36 Å), so the exchange interaction due to spin density on N 1 is stronger than that due to a same amount of spin density on C 1 . The resulting contribution from N 1 can thus be smaller or larger in absolute value than the contribution from C 1 . If it is smaller, the overall exchange interaction is still ferromagnetic, but weaker than it was before spin polarization. On the other hand, if it is larger, the overall exchange interaction turns from ferromagnetic into antiferromagnetic. In our RASSCF calculations we observe the first case.

Conclusion
We have presented results of a computational study of the intramolecular exchange coupling between Ln 3+ 4f electrons and the Pc 2 radical in [LnPc 2 ] 0 (Ln=Tb, Dy, Ho, and Er) molecules. We performed a series of state-averaged CASSCF and RASSCF calculations with and without SOC. When SOC is not considered, CASSCF calculations with minimum active space show that the coupling between lanthanides and the radical are all ferromagnetic, and that the magnitude of the exchange gap drops as the central metal goes from Tb to Er. On the other hand, inclusion of additional π-π * excitations via RASSCF calculations suggests a key role played by the polarized spin density on the nitrogen atoms, induced by the spin polarization effect on the Pc 2 radical. The negative spin density on the nitrogen atoms introduces an antiferromagnetic exchange pathway, weakening the overall ferromagnetic coupling strength between lanthanides and the radical.
The small ferromagnetic coupling calculated for [TbPc 2 ] 0 agrees with the latest experimental EPR evidence 23 but conflicts with the susceptibility measurements of Trojan et al. 19,20 Their data could only be explained by a large antiferromagnetic coupling, which our calculations do not support.