Explain the effects where the energy ordering between the #pi_(2p)# and #sigma_(2p)# MOs switches going from #"N"_2# to #"O"_2#?

The molecular orbitals of diatomic elements in the second period have orbital mixing effects, and from nitrogen to oxygen, the #sigma_(2p_z)# becomes lower in energy than the #pi_(2p_x)# and #pi_(2p_y)# orbitals since these effects have been decreasing across the period from left to right, and for nitrogen it is borderline.

The relevant orbitals are the #sigma_(g)(2s)# (the #sigma_(2s)#), the #sigma_(g)(2p)# (the #sigma_(2p_z)#), and the #pi_(u)(2p)# (the #pi_(2p_x)# and #pi_(2p_y)#).

Due to orbital mixing effects, which are most prevalent in #"Li"_2# and least prevalent in #"Ne"_2#:

• the #sigma_(g)(2s)# orbital is lower in energy than it would be without these effects.
• the #sigma_(g)(2p)# orbital is higher in energy than it would be without these effects.

These orbital mixing effects have a lesser and lesser effect as we go from left to right on the periodic table, so:

• the #sigma_(g)(2s)# orbital increases in energy (or decreases by less) as we move from left to right on the periodic table.
• the #sigma_(g)(2p)# orbital decreases in energy (or increases by less) as we move from left to right on the periodic table.

You can see that at nitrogen, the #sigma_(g)(2p)# orbital is slightly higher in energy than the #pi_(u)(2p)# orbitals, but for oxygen, the ordering switches.

It's because the orbital energy of the #sigma_(g)(2p)# as been increasing by less and less relative to without the orbital mixing effects, and going from nitrogen to oxygen, the effects are small enough that the #sigma_(g)(2p)# orbitals are overtaken by the #pi_(u)(2p)# orbitals in energy.

As for why the orbitals "mix", the quick reason is because they have the same "symmetry". It's a quantum mechanical phenomenon.