  Electron correlation effects play a key role in stabilising two-electron
atoms near the critical nuclear charge, representing the smallest charge
required to bind two electrons. However, deciphering the importance of these
effects relies on fully understanding the uncorrelated Hartree-Fock
description. Here, we investigate the properties of the ground state wave
function in the small nuclear charge limit using various symmetry-restricted
Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry
breaking occurs to give an unrestricted wave function that predicts the
ionisation of an electron. We also discover another critical nuclear charge
where the closed-shell electron density detaches from the nucleus, and identify
the importance of fractional spin errors and static correlation in this limit.
