Polyradical Character and Spin Frustration in Fullerene Molecules: An Ab Initio Non-Collinear Hartree–Fock Study

Most {\em ab initio} calculations on fullerene molecules have been carried out based on the paradigm of the H\"uckel model. This is consistent with the restricted nature of the independent-particle model underlying such calculations, even in single-reference-based correlated approaches. On the other hand, previous works on some of these molecules using model Hamiltonians have clearly indicated the importance of short-range inter-atomic spin-spin correlations. In this work, we consider {\em ab initio} non-collinear Hartree--Fock (HF) solutions for representative fullerene systems: the bowl, cage, ring, and pentagon isomers of C$_{20}$, and the larger C$_{30}$, C$_{36}$, C$_{60}$, C$_{70}$, and C$_{84}$ fullerene cages. In all cases but the ring we find that the HF minimum corresponds to a truly non-collinear solution with a torsional spin density wave. Optimized geometries at the generalized HF (GHF) level lead to fully symmetric structures, even in those cases where Jahn-Teller distortions have been previously considered. The nature of the GHF solutions is consistent with the $\pi$-electron space becoming polyradical in nature: each $p$-orbital remains effectively singly occupied. The spin frustration, induced by the pentagon rings in an otherwise anti-ferromagnetic background, is minimized at the HF level by aligning the spins in non-collinear arrangements. The long-range magnetic ordering observed is reminiscent of the character of broken symmetry HF solutions in polyacene systems.