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Methane Symmetry Operations

11.   Applications of the Td Symmetry Species

11.1 Dipole transition selection rules

All three laboratory-fixed components of the electric dipole-moment operator are of species A2. The molecule-fixed components are of species F2. Thus, an electric-dipole transition between a hyperfine level of overall species italic vr nGamma′ and one of overall species italic vr nGamma″ is rigorously forbidden unless italic vr nGamma′ × italic vr nGamma″ contains A2. The same transition is rovibrationally forbidden unless italic vrGamma′ × italic vrGamma″ also contains A2. It is vibrationally forbidden unless italic vGamma′ × italic vGamma″ contains F2.

Magnetic-dipole transitions are observed in molecular-beam studies of methane [42]. It can be shown that all three laboratory-fixed components of the magnetic dipole moment operator are of species A1. Thus, magnetic dipole transitions between hyperfine components are rigorously forbidden unless italic vr nGamma′ × italic vr nGamma″ contains A1.

Figure 5 illustrates a number of electric-dipole rovibrationally allowed transitions observed in methane. Solid vertical lines indicate strongly allowed vibration-rotation transitions of the italic v3 fundamental band [43-45]. Dashed lines indicate weakly allowed vibration-rotation transitions [46]. Dotted lines indicate very weakly allowed pure rotational transitions seen in double-resonance experiments [47-49].

The strong transition F1(2)-F2(2) nearly coincides with the 3.39 µm line of the He-Ne laser. Shimoda suggested [50] using this near coincidence and the Lamb-dip effect to achieve extreme stabilization [51-54] of the laser line. In such experiments Hall and Bordé [55] have resolved the hyperfine structure [25, 56] of this methane transition and have convincing line-shape evidence for the observation of photon-recoil effects [57].  

11.2 Perturbations and perturbation operators

The most fundamental selection rule concerns mixing and interactions evoked by the Hamiltonian operator among the functions of some basis set. Since the Hamiltonian is of species A1, only functions of the same species can mix or can perturb each other.

We now turn to two brief examples of the construction of individual interaction terms for the Hamiltonian operator. These constructions are best carried out using molecule-fixed components of the various vector operators, since molecule-fixed components are automatically invariant to those operations which correspond simply to rotating the molecule in space without permuting any identical particles and which are associated with changes in the m quantum number (see Sec. 15).

Consider first a vibration-rotation Coriolis operator which is to be bilinear in the (molecule-fixed) components of L and J. Since L and J both belong to the F1 representation, and since F1 × F1 contains the A1 representation only once, there is only one bilinear form allowed in the Hamiltonian. It can be seen from the matrices in Table 3 that JxLx + JyLy + JzLz is the desired operator.

Very similar considerations apply to the construction of the proton-spin - overall-rotation interaction operator [17], except that Table 19 contains two proton-spin vector operators belonging to the species F1. Thus there are two spin-rotation operators, having the forms: JxIx + JyIy + JzIz and Jx(I1y + I2y - I3y - I4y - I1z + I2z + I3z - I4z) + Jy(I1x + I2x - I3x - I4x + I1z - I2z + I3z - I4z) + Jz(- I1x + I2x + I3x - I4x + I1y - I2y + I3y - I4y), respectively.

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