Rotational energy levels oblate

Figure 5.6 Rotational energy levels for (a) a prolate and (b) an oblate symmetric rotor...

The rotational energy levels for a prolate and an oblate symmetric rotor are shown schematically in Figure 5.6. Although these present a much more complex picture than those for a linear molecule the fact that the selection mles... 

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... 

Figure 5.5 Rotational energy levels for oblate and prolate symmetric tops. Note that levels do not exist for K >J (dashed lines). For given J, the rotational energy is a decreasing (increasing) function of /T for oblate (prolate) tops.

Asymmetric tops have all three rotational constants unequal (A > B > C), Limiting cases are the prolate (B C) and oblate (B A) tops. Rotational energy levels are specified by three quantum numbers J, and K+ (for example 532,7n) which are sometimes written in a shorter, but equivalent form where r = K — 7 +i (in this case 5i,7 6)- The shorter nomenclature contains the same information since the sum of K and K+ may only adopt the values J or J+ 1. 

Figure 7-2. A rotational energy level diagram for an oblate symmetric top molecule is shown. Each column represents a K manifold of rotational J energy levels. In a rotational spectrum, allowed transitions occur within a particular K manifold because of the selection rule AK = 0.

When /a /b Ic, the molecule is an asymmetric rotor. There is then no closed form for the rotational energy eigenvalues but they can be found by interpolation between the energy levels of the prolate and oblate symmetric rotors with /a < /b = /c and / = /b < Ic-... 

For an oblate symmetrical top an identical expression results except that the unique axis is c, and thus A in Eq. (1.85) is replaced by C = h %n IcC), Since the rotation of a symmetric top molecule about its unique or symmetry axis results in no change in dipole moment, infrared radiation cannot change K which characterizes the angular momentum component about the unique axis. The selection rules are AK = 0 and = 1 for a symmetric top with a permanent dipole moment. When the energy difference between two successive energy levels (J + 1 is evaluated by subtracting Eq. (1.85) from the same equation where J + 1 has been substituted for we obtain... 

As apparent from Eqs. (29) and (31), the energy levels increase with K for a prolate rotor (A> B) and decrease with K for an oblate rotor (C < B). There are 7 -I-1 different rotational levels for each J value since the energy does not depend on the sign of K. The rotational levels for / 2 3 are illustrated in Fig. 7. Furthermore, in the absence of external fields each level is (27 - - l)-fold degenerate in the space orientation quantum number M. For absorption of radiation, the important selection rules are... 

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. 

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