In CH₃Cl for example, the main symmetry axis is the C – Cl axis.  We need two quantum numbers to describe the rotational motion with respect to I_A and I_C respectively.  Let J represent the total angular momentum of the molecule and K the angular momentum with respect to the C – Cl axis of the symmetric top.  J takes on integer values and K can not be greater than J (recall that m_l≤ ‌‌ l |  for orbital angular momentum).  The (2J + 1) “degeneracy” is expressed through the 2J + 1 values that K can take.

                                    K  =  J,  J – 1, …..  0,         -  (J – 1),  - J                                             (13.15)

The rotational energies of a symmetric top are given by

                                       (13.16)

The moments of intertia are related to B and A as

            and                                                               (13.17)

As the energy depends on K², energies for states with + K and – K are doubly degenerate. Thus there will be
J + 1 levels and (2J + 1) states for each values of J.

The selection rules for the symmetric top are,

ΔJ  =  ± 1 and ΔK = 0                                                                                                           (13.18)

It can be easily shown hat