SYMMETRIC TOP MOLECULE

Rotational energy level of this type of molecules are more complicated. Because of their symmetry their pure rotational spectra are relatively simple e.g CH₃F. For this type of molecule lB = I꜀ ≠ IA ; IA ≠ 0.

                          There are two directions of rotation in which the molecule absorbs or emits energy. Hence two quantum numbers are required to describe the degrees of rotation one for IA and one for IB or IC. However total angular momentum of the molecule is represented by single quantum number which is the sum of the separate angular momenta about the two different axes. This is usually chosen to be J. Angular momentum about the top axis is represented by K- i.e about C-F bond axis. The allowed values of J and K are zero or integral values. For a total momentum. J, K can take up values,

K = J, J-1, J-2,...........0.........-(J-1), - J with a total of 2J+1 values.

Schrodinger equation can be solved for rotational energy as, 

∈J,k = EJ,k/hc = BJ( J+1)+ (A-B) k² cm⁻¹........ (1)

Where B= h/8π² lAC

For all k>0, the rotational energy levels are degenerative. The selection rule is ∆J = ±1

and ∆K =0. Applying the selection rule, The wave number of the spectral line is given by

∈J +1, k − ∈J,k = V̂J,k =B(J+1) (J+2) +

(A−B) k² − [ BJ (J+1) + (A−B) k² ]

     = 2B (J+1) cm⁻¹  .............................(2)

Thus the spectrum is independent of k, and hence rotational changes about symmetry axis do not give rise to a rotational spectrum. The reason is that the rotation about the symmetry axis doesn't change the dipole moment and hence don't interact with radiation.

For a non-rigid rotor centrifugal force must be taken into account and the energy levels are given by

∈J,k = BJ(J+1) + (A−B) k² −DjJ² (J+1)² − DJk J(J+1) k² − DkK⁴ cm⁻¹  .................(3)

DJ, DJ,k and Dk are little correction terms for non- rigidity. The selection are same. Hence,

                 V̂ Jk = ∈J+1,k − ∈ Jk

Fitting the data from the spectrum above the equation, we can get the values of B, DJ and DJk.

     B= 0.851204 cm⁻¹

     DJ = 2.00* 10⁻⁶ cm⁻¹

     DJK = 1.47* 10⁻⁵ cm⁻¹

Spectra of molecule with different isotopes gives sufficient information to calculate all the bond lengths, force constants and bond angels of each bond of symmetric top molecule.

ASYMMETRIC TOP MOLECULES :-

                               The molecules have all the three moments of inertia different. No analytical equation is obtained for the rotational energies and their spectra are complex.