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.