Oblate top

In the symmetric top cases, Hrot can be expressed in terms of J2 and the angular momentum along the axis with the unique moment of inertia (denoted the a-axis for prolate tops and the c-axis of oblate tops) ... 

Wherever this is applicable in rotational fine structure analysis, the customary approach involves analyzing the spectra in terms of a near prolate or oblate top. Deviations from this approximation decrease with increasing quantum number K. For a more extensive discussion, for example of the subband structure in symmetric top spectra, the reader is referred to Allen and Cross (1963). 

Many nonlinear molecules can be treated as symmetric top rotors in which two of the moments of inertia are equal. The moment of inertia about the synunetiy axis is 7, while the two other moments of inertia are 7 = ly A symmetric top can be visualized as a rotating cylinder. For a given J, the cylinder can rotate in a total of 27 -I- 1 orientations, each with a different K quantum number which determines its projection along the symmetry axis. Figure 7.9 shows the case of prolate and oblate tops rotating with K J and K = 0. 

Symmetric Top, Ia - Iq (prolate top) or Ia=Ibrotational spectrum is still degenerate in the nij quantum number however, part of the k degeneracy is lift. Only those levels with k =0 are now doubly degenerate (in k). The rotational spectrom of symmetric top molecules (such as NH3, CgHg) can be written as... 

In order to determine the band profiles, the shape of the molecule has to be considered [7], As is well known, in symmetric prolate or oblate tops, the... 

Figure 5.3 Examples of a prolate symmetric top, 2-butyne (left) and an oblate symmetric top, benzene (right). By convention, the figure axis is labeled the a axis and the c axis in prolate and oblate tops, respectively.

A schematic energy level diagram is given for the oblate top in Fig. 5.5. 

Figure 5.6 Rotational motion in an oblate top for (a) K =J and (b) /f = 0. According to Eq. 5.25, the rotational energy for given J is smaller in case (a) than in case (b). This is in agreement with the classical result, where the rotational energy J lll depends on the total angular momentum J and the moment of Inertia / about the axis of rotation / is clearly larger in case (a) than in case (b).

In a prolate symmetric top, the a axis rather than the c axis becomes the figure axis. The body-fixed a, b, c axes depicted in Fig. 5.4 are replaced by the b, c, and a axes, respectively. The mutually commuting set of observables becomes J, Ja and J. With these modifications, the oblate top eigenvalue equations and wave functions (5.21)-(5.24) become applicable to the prolate top as well. The pertinent rotational energies in cm are... 

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