= Solution
Both Be$_2$'s occupied bonding and antibonding 2s levels cancel in the simplest diagram. <Beryllium>'s relatively accessible empty 2p orbitals allow mixing and correlated configurations to change that cancellation and produce a \b[weak bond]. An s-only single-configuration picture is therefore insufficient; spectroscopy and accurate correlated calculations establish its binding.<Beryllium dimer caught in the act of bonding by Jeremy M. Merritt and colleagues>{ref}
For He$_2$, $(\sigma_{1s})^2(\sigma_{1s}^*)^2$ similarly gives zero elementary <covalent bond> order, but the empty-shell excitation gap is much larger and there is no comparable low-energy 2s/2p near-degeneracy. Thus it has \b[no ordinary stable <covalent bond>].
The literal claim that He$_2$ does not exist needs qualification: an extraordinarily weak, cold van der Waals <helium dimer> has been observed. That does not contradict the zero covalent-bond-order diagram.<The weakest bond: experimental observation of helium dimer by Fei Luo and colleagues>{ref}
The <beryllium dimer> and <helium dimer> illustrate two physically different ways that the elementary bond-order picture can miss weak binding.
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