## Methane Symmetry Operations

#### 16.2 Coupling of vibrational and rotational basis functions

Let us now apply (eq. 59) to two examples.
First consider the coupling of vibrational and rotational angular momenta
already discussed in Section 14. The functions
| *k J m*⟩
are eigenfunctions of the operators **J**^{2},
*J*_{z }, and the functions | _{L} ⟩ are eigenfunctions of the
operators _{s}^{2},
_{sz }, with
*s* = 3 or 4. Furthermore, these two sets of eigenfunctions
transform, as given in (eq. 33) and
(eq. 48), according to the irreducible
representations *D*^{(J)} and *D*^{(L)}
of the continuous three-dimensional rotation group (or according to
*D*_{g}^{(J)} and
*D*_{u}^{(L)}, for *L* = 1, of the
continuous three-dimensional rotation-reflection group).
We wish to construct eigenfunctions of the operators
**J**^{2}, _{s}^{2},
**R**^{2} ≡ (**J** +
_{s})^{2} and
*R*_{z }. According to
(eq. 59) we have

Eigenfunctions on the left with fixed *J*, *m*,
, and *R* transform according
to the irreducible representation *D*^{(R)} of the
continuous three-dimensional rotation group when both rotational and
vibrational variables are subjected to "point-group-type" rotations.

Two interesting points arise here. First, the quantum number *R*
represents an approximation to the purely rotational angular momentum, since
*R* is the difference between the total angular momentum and
1/ζ_{s} times the
vibrational angular momentum.

The second point is something of an anomaly. The quantum number
determines the amount of
*vibrational* angular momentum and the transformation properties of the
*vibrational* wave functions. On the other hand, the quantum number
*J* determines the amount of *total* (vibration-rotation) angular
momentum and the transformation properties of the *rotational* wave
functions, while the quantum number *R* gives an approximation to the
amount of *rotational* angular momentum and determines the transformation
properties of the *total* (vibration-rotation) wave functions.

Since the eigenfunctions | *J m* *R k*_{R} ⟩ with
= 1 transform according
to the representation *D*_{u}^{(R)} of the
continuous three-dimensional rotation-reflection group,
Table 14 immediately gives the
*T*_{d} symmetry species which can occur for given values of
*J*, = 1 and
*R*. The three groupings of *υ*_{3} = 1 symmetry species in
Figure 5 correspond to *T*_{d}
species for *D*_{u}^{(R)} with
*R* = 7, 6, 5.

#### 16.3 Coupling of nuclear spin basis functions

Consider now the introduction of nuclear spin. We can couple the
laboratory-fixed projections *m*_{I} of the nuclear spin functions
| *Γ**I m*_{I} ⟩ with the laboratory-fixed
projections *m* of the rovibrational functions
| *J m* *R k*_{R} ⟩ to obtain
eigenfunctions of the operators **F**^{2} ≡
(**J** + **I**)^{2} and
*F*_{Z}. Since the laboratory-fixed components of both
**J** and **I** commute with the ordinary
[29] sign of *i*, atomic coupling formalism can
be used immediately. We obtain

Furthermore, since the coupling in (eq. 61) involves projections along a
laboratory-fixed axis, the continuous three-dimensional rotation-reflection
group is the appropriate symmetry group to consider, and no reduction of
*D*^{(F)} into species of the group *T*_{d} is
to be performed. Also, the coupling in (eq. 61) is not a recoupling in the
usual sense; summation is over an as yet unused projection *m* of one of
the already coupled angular momenta *J*, a situation which does not arise
in atomic problems.

There remain some considerations associated with the point group
*T*_{d} which have yet to be taken into account. From an
examination of transformation properties under point-group rotations and under
laboratory-fixed rotations, it can be shown that quantum numbers of the
functions |*k* *J* *m*⟩ and
|*Γ* *I* *m*_{I }⟩ are analogous
in pairs: *J* and *I* determine the magnitude of angular momentum
involved; *m* and *m*_{I} determine its projection along the
laboratory-fixed *Z* axis as well as transformation properties of the
functions under laboratory-fixed rotations; and *k* and
*Γ* determine transformation
properties under point-group rotations. Unfortunately, as can be seen by
evaluating the appropriate commutation relations, it is not possible
simultaneously to quantize nuclear spin projections along one laboratory-fixed
axis *and* one molecule-fixed axis, as is done for the total angular
momentum **J**. Thus, the pairwise analogy is not quite exact.

In any case, the nuclear spin functions transform as indicated in
Table 15 and
Table 16 under the *T*_{d}
point-group operations. The rovibrational functions transform as indicated in
(eq. 33) and
Table 13, with *J* and *k*
replaced by *R* and *k*_{R}, under these same operations. It
is necessary to couple the value of *k*_{R} with the symmetry
species of the nuclear spin functions (e.g., *Γ* = *E*_{a }, *E*_{b },
*F*_{2x }, etc.) to obtain overall wave functions which
transform according to irreducible representations of the point group
*T*_{d} and which obey the correct statistics (see
Section 9.2).

The coupling of *k*_{R} with the nuclear spin species is a true
recoupling, since *k*_{R} is itself a quantum number obtained by
coupling *k* and _{L}. This recoupling can be carried out using
atomic formalism when the symmetry species of the nuclear spin functions of
given total spin *I* correspond to the reduction of one of the species
*D*_{g ,u}^{(j)} of the three-dimensional
rotation-reflection group into species of *T*_{d} (e.g.,
*F*_{2} → *D*_{u}^{(1)}, but
*E*
*D*_{g ,u}^{(j)} for CH_{4}). Such
recoupling is probably best carried out in general, however, using irreducible
tensor techniques specifically adapted for tetrahedral molecules
[63]. Here, of course (as indeed at any point along
the way), a reader doing numerical calculations can retreat to the point group
*D*_{2d }, where nuclear-spin coupling becomes almost
trivial [25].