Rotational energy levels prolate

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... 

The energy and symmetry properties of rotational levels have been discussed comprehensively by Herzberg (1945). The principles are unchanged as between the different classes, although the details vary somewhat from one class to another. Here we shall consider only the case of a near-prolate asymmetric top (lcxlb >la), a class which contains formaldehyde, propynal, and (raws-bent acetylene among other examples. In first approximation the rotational energy levels are expressed in terms of the rotational constants A, B, and C, where... 

Before discussing the spherical rotor, it is appropriate to focus on the rotational energy levels for symmetric rotors (see also Tab. 4.3-2). For the rigid prolate top, the rotational energy is given 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.

This variety in rotational selection rules, coupled with our natural endowment of molecules with diverse rotational constants, leads to wide variations in the rotational fine structure exhibited by symmetric and near-symmetric tops. For definiteness, we consider a prolate symmetric top whose rotational energy levels are given in Eq. 5.26. Rotational lines will be found at the frequencies... 

Using perturbation theory, Watson" has derived an expression, similar to eqn (20.31) but with rotation included, for a polyatomic molecule s vibration/ rotation energy levels. The analytic form for a prolate symmetric top has been given." This approach assumes the molecule has good vibration/rotation quantum numbers and, if this is not the case or if more accurate values for the energy levels are required, the variational method may also be used to determine vibrational/rotational energy levels. For this approach the wavefunction for a given vibration-rotation level is written as a linear combination of basis function j/iRjK -... 

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. 

All of the experimental methods discussed above are used to study the rotational spectra of dimers. The main object is to extract rotational constants for a number of isotopically substituted analogs, and from that data, to calculate the geometry of the dimer. The process is aided by the apparent simplicity of the rotational spectra. Many of the dimers discussed below are either linear molecules or slightly asymmetric rotors in the prolate limit. The rorational energy levels are then given by... 

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-... 

The aeeeptor switching splitting leads to a set of energy levels with symmetries Ai/Ei/Bi and another set of energy levels with A2/E2/B2. All transitions were fit to the foUowing rotational energy expressions appropriate for near prolate tops [03Keul] ... 

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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