If the vibrational frequency is ω0 the excitation energy of the first vibrational level will be hωo. Thus if there are J rotational levels contained in the band between the ground state and the first vibrational excitation, then
where as stated in the problem we have ignored any coupling between the two. For HF molecule
For J = 14 , J (J + 1 ) = 210 . For J - 13, J (J + 1) = 182. Thus there lie 13 levels between the ground state and the first vibrational excitation.