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Most {\em ab initio} calculations on fullerene molecules have been carriedout based on the paradigm of the H\"uckel model. This is consistent with therestricted nature of the independent-particle model underlying suchcalculations, even in single-reference-based correlated approaches. On theother hand, previous works on some of these molecules using model Hamiltonianshave clearly indicated the importance of short-range inter-atomic spin-spincorrelations. In this work, we consider {\em ab initio} non-collinearHartree--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 butthe ring we find that the HF minimum corresponds to a truly non-collinearsolution with a torsional spin density wave. Optimized geometries at thegeneralized HF (GHF) level lead to fully symmetric structures, even in thosecases where Jahn-Teller distortions have been previously considered. The natureof the GHF solutions is consistent with the $\pi$-electron space becomingpolyradical in nature: each $p$-orbital remains effectively singly occupied.The spin frustration, induced by the pentagon rings in an otherwiseanti-ferromagnetic background, is minimized at the HF level by aligning thespins in non-collinear arrangements. The long-range magnetic ordering observedis reminiscent of the character of broken symmetry HF solutions in polyacenesystems.

Polyradical character and spin frustration in fullerene molecules: An ab initio non-collinear Hartree--Fock study