To determine which molecule does not exist, we need to consider the molecular orbital theory.$\\$Molecular orbital theory helps us understand the stability of diatomic molecules by calculating the bond order.$\\$The bond order is given by the formula:$\\$Bond order$= \frac{N_b - N_a}{2} \\$where $N_b$ is the number of electrons in bonding orbitals and $N_a$ is the number of electrons in antibonding orbitals.$\\$Let's analyze each option:$\\$Option 1: $$Li$_2 \\$• Li has 3 electrons. In $$Li$_2,$ there are 6 electrons in total.$\\$• Electron configuration: $\sigma_{1s}^2 \sigma_{1s}^*{}^2 \sigma_{2s}^2 \\$• Bond order$= \frac{4 - 2}{2} = 1 \\$• Since the bond order is positive, $$Li$_2$ exists.$\\$Option 2: $$C$_2 \\$• C has 6 electrons. In $$C$_2,$ there are 12 electrons in total.$\\$• Electron configuration: $\sigma_{1s}^2 \sigma_{1s}^*{}^2 \sigma_{2s}^2 \sigma_{2s}^*{}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \\$• Bond order$= \frac{8 - 4}{2} = 2 \\$• Since the bond order is positive, $$C$_2$ exists.$\\$Option 3: $$O$_2 \\$• O has 8 electrons. In $$O$_2,$ there are 16 electrons in total.$\\$• Electron configuration: $\sigma_{1s}^2 \sigma_{1s}^*{}^2 \sigma_{2s}^2 \sigma_{2s}^*{}^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \pi_{2p_x}^*{}^1 \pi_{2p_y}^*{}^1 \\$• Bond order$= \frac{10 - 6}{2} = 2 \\$• Since the bond order is positive, $$O$_2$ exists.$\\$Option 4: $$He$_2 \\$• He has 2 electrons. In $$He$_2,$ there are 4 electrons in total.$\\$• Electron configuration: $\sigma_{1s}^2 \sigma_{1s}^*{}^2 \\$• Bond order$= \frac{2 - 2}{2} = 0 \\$• Since the bond order is zero, $$He$_2$ does not exist.$\\$Therefore, the molecule that does not exist is $$He$_2$, which corresponds to Option 4.