To determine which diatomic molecular species has only $\pi$ bonds according to Molecular Orbital Theory, we need to analyze the molecular orbital configurations of each species.$\\$1. $O_2: \\$The molecular orbital configuration for $O_2$ is:$\\ ( \sigma_{1s})^2 ( \sigma^*_{1s})^2 ( \sigma_{2s})^2 ( \sigma^*_{2s})^2 ( \sigma_{2p_z})^2 ( \pi_{2p_x})^2 ( \pi_{2p_y})^2 ( \pi^*_{2p_x})^1 ( \pi^*_{2p_y})^1 \\$O$_2$ has both $\sigma$ and $\pi$ bonds.$\\$2. $N_2: \\$The molecular orbital configuration for $N_2$ is:$\\ ( \sigma_{1s})^2 ( \sigma^*_{1s})^2 ( \sigma_{2s})^2 ( \sigma^*_{2s})^2 ( \pi_{2p_x})^2 ( \pi_{2p_y})^2 ( \sigma_{2p_z})^2 \\$N$_2$ has both $\sigma$ and $\pi$ bonds.$\\$3. $C_2: \\$The molecular orbital configuration for $C_2$ is:$\\ ( \sigma_{1s})^2 ( \sigma^*_{1s})^2 ( \sigma_{2s})^2 ( \sigma^*_{2s})^2 ( \pi_{2p_x})^2 ( \pi_{2p_y})^2 \\$C$_2$ has only $\pi$ bonds and no $\sigma$ bonds in the $2p$ orbitals.$\\$4. $Be_2: \\$The molecular orbital configuration for $Be_2$ is:$\\ ( \sigma_{1s})^2 ( \sigma^*_{1s})^2 ( \sigma_{2s})^2 ( \sigma^*_{2s})^2 \\$Be$_2$ does not have any $\pi$ bonds.$\\$Therefore, the correct option is $3$ (C$_2$), which has only $\pi$ bonds.