According to molecular orbital theory, the species among the following that does not exist is:

According to Molecular Orbital Theory, a molecule exists only if its bond order is greater than zero. Bond order is calculated as $$\text{B.O.} = \frac{N_b - N_a}{2}$$, where $$N_b$$ is the number of bonding electrons and $$N_a$$ is the number of antibonding electrons.

For $$He_2^-$$ (5 electrons): The configuration is $$(\sigma 1s)^2(\sigma^* 1s)^2(\sigma 2s)^1$$. Bond order $$= \frac{3 - 2}{2} = 0.5$$. This species can exist.

For $$O_2^{2-}$$ (18 electrons): The configuration fills up to $$(\pi^* 2p)^4$$. Bond order $$= \frac{10 - 8}{2} = 1$$. This species can exist.

For $$He_2^+$$ (3 electrons): The configuration is $$(\sigma 1s)^2(\sigma^* 1s)^1$$. Bond order $$= \frac{2 - 1}{2} = 0.5$$. This species can exist.

For $$Be_2$$ (8 electrons): The configuration is $$(\sigma 1s)^2(\sigma^* 1s)^2(\sigma 2s)^2(\sigma^* 2s)^2$$. Bond order $$= \frac{4 - 4}{2} = 0$$. With zero bond order, this molecule does not exist.

Therefore, the species that does not exist is $$Be_2$$, which corresponds to option (4).