  We present a thorough analysis of symmetry breaking observed in Hartree-Fock
(HF) solutions of fullerenes C$_{60}$, C$_{36}$, and C$_{20}$ in order to
characterize the nature of electron correlation in them. Our analysis is based
on (1) the critical regularization strength to restore symmetry breaking in the
recently developed regularized orbital optimized second-order M{\o}ller-Plesset
perturbation theory ($\kappa$-OOMP2), (2) singlet-triplet gaps from various MP2
methods, and (3) natural orbital occupation numbers from restricted
coupled-cluster with singles and doubles (RCCSD) and coupled-cluster valence
bond with singles and doubles (CCVB-SD). Based on these three independent
probes, we conclude that C$_{36}$ (D$_\text{6h}$) exhibits genuine strong
correlation and symmetry breaking whereas C$_{60}$ exhibits {\it artificial} HF
symmetry breaking and is not strongly correlated. Investigating the critical
regularization strength, we discuss strong correlation in C$_{20}$ at the
Jahn-Teller distorted geometries (C$_\text{2h}$, D$_\text{2h}$, C$_\text{i}$,
and D$_\text{3h}$) and the I$_\text{h}$ geometry. Only C$_{20}$ (I$_\text{h}$)
was found to be strongly correlated while others exhibit {\it artificial} HF
symmetry breaking. This analysis highlights a useful feature of the recommended
$\kappa$-OOMP2 method. It is an electronic structure method that describes
dynamic correlation, and attenuates strong correlation in MP2 towards zero by
regularization. Therefore, $\kappa$-OOMP2 will exhibit symmetry breaking in its
reference determinant only when correlation is strong (i.e., essential symmetry
breaking). Artificial symmetry breaking (arising in HF due to neglect of
dynamic correlation) is thus removed in $\kappa$-OOMP2.
