## Methane Symmetry Operations

### 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 *A*_{2}. The molecule-fixed components are of
species *F*_{2}. Thus, an electric-dipole transition between a
hyperfine level of overall species
^{υr n}*Γ*′ and one of overall species
^{υr n}*Γ*″ is rigorously forbidden unless
^{υr n}*Γ*′ × ^{υr n}*Γ*″ contains *A*_{2}.
The same transition is rovibrationally forbidden unless
^{υr}*Γ*′ × ^{υr}*Γ*″ also contains
*A*_{2}. It is vibrationally forbidden unless
^{υ}*Γ*′ × ^{υ}*Γ*″ contains *F*_{2}.
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
*A*_{1}. Thus, magnetic dipole transitions between hyperfine
components are rigorously forbidden unless
^{υr n}*Γ*′ × ^{υr n}*Γ*″ contains *A*_{1}.

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
*υ*_{3} 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
*F*_{1}^{(2)}-*F*_{2}^{(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 *A*_{1}, 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
*F*_{1} representation, and since
*F*_{1} × *F*_{1} contains the
*A*_{1} 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*_{x}*L*_{x} +
*J*_{y}*L*_{y} +
*J*_{z}*L*_{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 *F*_{1}. Thus there are
two spin-rotation operators, having the forms:
*J*_{x}I_{x} +
*J*_{y}I_{y} +
*J*_{z}I_{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.