Methane Symmetry Operations - Td Symmetry Species

11.   Applications of the T_d 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 ^υr nΓ′ and one of overall species ^υr nΓ″ is rigorously forbidden unless ^υr nΓ′ × ^υr nΓ″ contains A₂. The same transition is rovibrationally forbidden unless ^υrΓ′ × ^υrΓ″ also contains A₂. It is vibrationally forbidden unless ^υΓ′ × ^υΓ″ 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 ^υr nΓ′ × ^υr nΓ″ 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 υ₃ 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 J_xL_x + J_yL_y + J_zL_z 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: J_xI_x + J_yI_y + J_zI_z and J_x(I_1y + I_2y - I_3y - I_4y - I_1z + I_2z + I_3z - I_4z) + J_y(I_1x + I_2x - I_3x - I_4x + I_1z - I_2z + I_3z - I_4z) + J_z(- I_1x + I_2x + I_3x - I_4x + I_1y - I_2y + I_3y - I_4y), respectively.