 Solution

N${}_2$: Homo ${\sigma }_{2p_z}$; LUMO the degenerate ${\pi }_{2p}^{\ast }$ pair.

O${}_2$: the two ${\pi }_{2p}^{\ast }$ orbitals are the highest occupied, singly occupied spatial orbitals. The next completely empty spatial orbital is ${\sigma }_{2p_z}^{\ast }$. An added electron can instead occupy the opposite-spin vacancy in a singly occupied pi-star orbital; this distinction matters if “LUMO” is being used for an electron-addition spin orbital.

Li${}_2$: Homo ${\sigma }_{2s}$; next empty sigma orbital conventionally denoted ${\sigma }_{2s}^{\ast }$ in this elementary diagram. Be${}_2$: Homo ${\sigma }_{2s}^{\ast }$; the low empty 2p manifold supplies the LUMO candidates, drawn as ${\pi }_{2p}$ in the stated light-diatomic ordering. Mixing makes these labels descriptions of dominant atomic-orbital character, not strictly unmixed atomic functions.

This distinction is the open-shell frontier orbital convention.