Symmetric top oblate

Molecules for which two of the three principal moments of inertia are equal are called symmetric tops. Those for which the unique moment of inertia is smaller than the other two are termed prolate symmetric tops if the unique moment of inertia is larger than the others, the molecule is an oblate symmetric top. 

Fig. 53 Rotational levels for a prolate symmetric top with A =4.55 and for an oblate symmetric top with B=4.5C. Note the accidental degeneracy of the 7=2, Af 2, and 7=4, K =0 prolate-top levels.

Rotors are classified as follows If all three principal moments of inertia are equal, we have a spherical top. If two principal moments are equal, we have a symmetric top. A prolate symmetric top has Ib = Ic and the a axis (the axis of smallest principal moment) is the symmetry axis an oblate symmetric top has la = lb and the c axis is the symmetry axis. A football (American, not British) is a prolate symmetric top a discus is an oblate symmetric top. If all three principal moments are unequal, we have an asymmetric top. Thus... 

For a symmetric top, two of the three rotational constants are equal. Either B = A and we have an oblate symmetric top ... 

Asymmetric top rotational states are labelled by the value of J (or N if S 0), with subscripts Ka,Kc, where the latter correlate with the K = k quantum number about the a and c axes in the prolate and oblate symmetric top limits respectively. 

The c-type transitions give rise to perpendicular bands in the prolate symmetric top limit and to parallel bands in the oblate symmetric top limit. Selection rules are then... 

A succession of levels like those of a linear molecule can be calculated for each quantum number K, which in this case describes the quantized component of the angular momentum about the unique a-axis. K cannot exceed 7, the quantum number for the total angular momentum, i.e., K = 0, 1,... dz7. For an oblate symmetric top the rotational constant A j has to be replaced by Q ]. In relation to the case of A" = 0, other K quantum numbers allowed will thus result in lower energies Ejk, which is in contrast to the prolate top with a positive term of (A[ j - 6 ]). Evidently, all rotational levels with 0 are doubly degenerate. It should be noted that each level still possesses an M-degeneracy of (27 -f 1) as discussed in connection with the linear molecule. This is due to space quantization. 

For a nonrigid symmetric top molecule there are three different centrifugal distortion constants 1, D", and ). The rotational energy levels of the vibrational level v of an oblate symmetric top can be represented by... 

As an example consider a planar, near oblate symmetric top molecule. If small amplitude vibrations are assumed, we may expect that the ju-tensor elements are of the order of the reciprocal principal moments of inertia. However, for a planar configuration, withIc-Ia+Ib, it is easily seen that the generally smallest element, ncc, may reach extreme values when evaluated in a PAS using Eq. (2.63). A numerical example illustrates this ... 

Fig. III.9. Low J section of the Hamiltonian matrix of an asymmetric top molecule such as for instance ethyleneoxide in the absence of exterior fields. The matrix is set up in the eigenfunction basis of the limiting oblate symmetric top, =...

In Table III.3 we have introduced the J,K-K+) notation for the rotational states instead of the J,r) notation which was used up until now in this paper. K- and K+ are the fC-quantum numbers of the limiting prolate and oblate symmetric tops respectively as is illustrated in Fig. III.10. 

Spectroscopic studies in the gas phase and very high level ab initio calculations indicate that this species is an oblate symmetric top belonging to the molecular point group (Fig. 3). The experimentally estimated C-H distance is 1.087 A, the computed value (1.089 A) being in excellent agreement. 

Figure 7.9 Prolate and oblate symmetric top molecules represented as cylinders. The rotational motion due to / and K are indicated.

The molecule PH3 (C3V symmetry) is an oblate symmetric top (Crotational constants C (refers to rotation around the C3 axis) and B (perpendicular to C3). Since the permanent electric dipole moment is pointed parallel to the C3 axis, only pure rotational transitions with the selection rule AK=0 are allowed (K is the quantum number of the component about the C3 axis of the total angular momentum J). Their analysis leads to the parameters B, Dj, Djk, and Hjk. From the perturbation-allowed transitions AK= 3n (n=1,2,...), which become weakly allowed by centrifugal distortion effects (inducing a small dipole moment of about 8x10 D perpendicular to the C3 axis [1, 2, 3]), the K-related constants (C, Dk, Hk) were obtained see, e.g. [1, 3, 4]. 

Q-branch transitions (AJ = 0) and some R-branch transitions (AJ = +1) were identified in the rotational spectra of PHDg (Cs symmetry oblate asymmetric top) and PHJD (Csi prolate asymmetric top). Transitions, number of lines (n), and the corresponding frequency range are given in Table 15. The rotational levels are denoted by Jk KcO i the table J, Kg, K ) Kg and K are the K values for the given level for the limiting cases of prolate and oblate symmetric top, respectively [16, 17]. 

A table of the microwave spectra of triatomic molecules [1] contains 139 lines for OF2 with frequencies ranging from 8299.51 to 59846.20 MHz, which were taken from the literature, and, in an appendix, 26 additional lines from 84401.810 to 95998.940 MHz and one line at 7803.49 MHz (transitions between rotational levels Jk-,k, with J = total angular momentum quantum number excluding nuclear spin and K", K" = projections of J on the symmetry axis of the limiting prolate or oblate symmetric top). Eight of these known rotational transitions and seven new transitions, belonging to the Vi =1 and Vg = 2 vibrationally excited states, were more recently observed by IR-microwave (MW) double resonance or IR-MW-MW triple resonance [12]. 

NF3 is an oblate symmetric top with rotational constants A=B>C. Equilibrium and effective rotational constants, Bg and Bq, rotation-vibration constants, af with i = 1 to 4, and centrifugal distortion constants, Dj and Djk, are presented in Table 2, p. 190. Values are from rotational microwave, rotational IR, or vibrational IR spectra (denoted as rot. MW, rot. IR, vibr. IR in Table 2). 

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.

In the oblate symmetric top, the rotational Hamiltonian is given by Eq. 5.10. The c axis is denoted the figure axis. According to the commutation rules obtained in Section 5.3, one possible commuting set of observables is J, J, and J. It is then possible to formulate rotational states JKM which simultaneously obey the eigenvalue equations... 

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