Spherical rotor

Linear, symmetric rotor, spherical rotor and asymmetric rotor molecules... 

A spherical rotor has all three principal moments of inertia equal ... 

We tend to think of a spherical rotor molecule, such as methane (see Figure 4.12a), as having no permanent dipole moment and, therefore, no infrared, millimetre wave or microwave... 

Neglecting centrifugal distortion, fhe rofafion term values for a spherical rotor are given by... 

All regular fefrahedral molecules, which belong to fhe poinf group (Section 4.2.8), may show such a rofational spectrum. However, those spherical rotors that are regular octahedral molecules and that belong to the Of, point group (Section 4.2.9) do not show any such... 

For a spherical rotor belonging to the octahedral Of, point group, Table A.43 in Appendix A, in conjunction with the vibrational selection rules of Equation (6.56), show that the only allowed transitions are... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

Squared Fluctuation Functions F2(f, /), Expressed in Terms of Autocorrelation Functions p(/,/) and the Limiting Forms Taken by the Scaled rms Fluctuation Functions (/(/,/) at Short and Long Times, for the Functions /Shown and the Following Systems . (A) Harmonically bound particle. (B) One-dimensional particle in a box. (C) Plane rotor. (D) Spherical rotor. The limiting behaviour at short times is given in terms of e = 1 — p. 

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... 

For a spherical rotor, the three moments of inertia are equal Ia = Ib = Ic = I-Examples are CH4 and SFg. The energy simplifies to J /2I and the quantum-mechanical Hamiltonian is given by... 

Derive a formula for the most highly populated level Jmax of a spherical rotor. Determine its value for CH4 at 300 K given that B = 5.24 cm . ... 

For a spherical rotor, the degeneracy increases to gy = (27+1) the Boltzmann distribution for rotational levels is then... 

---