The octet rule could satisfactorily explain how and why the atoms of the second period formed covalent bonds by sharing. However, it could not explain why certain atoms like Boron or Phosphorous did not follow the octet rule or why the Nitrogen atom in NO has an unpaired electron.
Incomplete octet
Beryllium has two, and Boron, or Aluminium, has three valence electrons. In compounds like BeCl2 or BCl3, the central Boron atom doesn’t complete its octet yet form covalent bonds, which the octet rule couldn’t explain.
So, the octet rule is unsatisfactory in explaining covalent bonding in atoms with less than four valence electrons.
Expanded Octet
Elements of the third period and above have d-orbitals that take part in bond-formation reactions. This allows the atom to incorporate a higher number of orbitals, leading to such atoms forming a greater number of bonds.
Phosphorus can form five bonds as PF5, and Sulphur can form six bonds as SF6, thereby expanding the octet to ten and twelve.
Odd electron
According to the octet rule, eight electrons and a complete octet in the outermost shell are why atoms form covalent bonds. However, there exist molecules that have an odd electron in their shell. The odd electrons are unlike the lone pair or bond pairs and are commonly called free radicals (denoted with a dot . over the atom).
For example, nitric oxide (.NO) and nitrogen dioxide (.NO2) have odd electrons.
The odd electron-containing atoms are reactive species, unstable, and some may largely exist as dimers.
Related Reading- The Octet Rule