Methane Symmetry Operations
7. Symmetry Properties of Rotational Wave functions and Direction
Cosines
It is in the determination of symmetry properties of functions of the Eulerian
angles, and in particular in the question of how to apply sensereversing
pointgroup operations to these functions, that the principal differences arise
in grouptheoretical discussions of methane. The treatment given here follows
from the discussion of Section 4. The two other
most commonly followed treatments, due to Jahn
[5−8] and MoretBailly
[9−11], respectively, will be discussed briefly
in Section 12.
7.1 Symmetrictop rotational basis functions
The moleculefixed components
J_{x},J_{y},J_{z} of the total
angular momentum operator can be written as [4]
where p_{χ} = −i(∂/∂_{χ}), etc.
[4, 28]. Since the
transformation from moleculefixed to laboratoryfixed vector components takes
place via the matrix S in (eq. 10),
the laboratoryfixed components
J_{X},J_{Y},J_{Z} of the total
angular momentum operator can be obtained from (eq. 23) and
Thus,
It can be shown, by direct application of the differential operators, that
symmetric top rotational basis functions kJm defined in terms of Wigner's
functions
(equation (15.27) of [22]) as
are eigenfunctions of the operators J_{z} and
J_{Z} belonging to the eigenvalues
k and
m, respectively, and give rise
to real positive matrix elements for the laboratoryfixed
[29] (J_{X} ± iJ_{Y}) and
moleculefixed [30] (J_{x} iJ_{y}) ladder
operators. Substitution of the Eulerian angle transformations of
Table 5 in the function
kJm
leads to the transformations shown in Table 11.
Table 11. Transformation of the symmetric top function
kJm
under the symmetry operations of the D_{2d} subgroup
given in Table 5

E kJm⟩ = 
kJm⟩ 

C_{2}(x) kJm⟩ = 
(−1)^{J} −kJm⟩ 
S_{4}(z) kJm⟩ = 
(−i)^{k} kJm⟩ 
C_{2}(y) kJm⟩ = 
(−1)^{J−k} −kJm⟩ 
S_{4}^{3}(z) kJm⟩ = 
(+i)^{k} kJm⟩ 
σ_{d}(110) kJm⟩ = 
(−1)^{J} (−i)^{k}
−kJm⟩ 
C_{2}(z) kJm⟩ = 
(−1)^{k} kJm⟩ 
σ_{d}(−110) kJm⟩ = 
(−1)^{J} (+i)^{k}
−kJm⟩ 

If we further take sums and differences for K =  k 
≥ 1,
we find symmetry species for these functions under the
D_{2d} symmetry operations as given in Table 12.
Table 12. Symmetry species in D_{2d} of the Wang sum
and difference rotational functions specified in (eq. 27)


J = even


J = odd


0Jm⟩^{ } 
K^{+}Jm⟩ 
−i K^{+}Jm⟩^{ } 
0Jm⟩^{ } 
K^{+}Jm⟩ 
−i K^{−}Jm⟩^{ } 

K = 0_{ } 
A_{1} 


A_{2} 


K = 1 mod 4 
E_{x} 
E_{y} 
E_{y} 
− E_{x} 
K = 2 mod 4 
B_{1} 
B_{2} 
B_{2} 
B_{1} 
K = 3 mod 4 
E_{x} 
− E_{y} 
E_{y} 
+ E_{x} 
K = 4 mod 4 
A_{1} 
A_{2} 
A_{2} 
A_{1} 

As mentioned earlier, functions belonging to a single symmetry species of the
full point group T_{d} are rather less convenient to write down.
In fact, it is necessary to introduce linear combinations of the symmetric top
functions much more complicated than sums and differences
[31]. These will not be given here, though
transformation properties of the kJm under operations of the full point group T_{d}
are given in Section 8.
7.2 Direction cosines
The discussion of transformation properties and symmetry species is rather
simple for the direction cosines. These quantities are just the nine elements
of the 3 × 3
rotation matrix S(χθφ), and
transformations of the Eulerian angles were in fact defined originally in
Section 4.2 and
Section 4.3 to insure certain transformation
properties of the direction cosines themselves. Equations
(eq. 14) and
(eq. 18) prescribe the transformations of
the direction cosine matrix when
χ_{new},
θ_{new},
φ_{new} are substituted for
χ,θ,φ. Since the matrices M
in (eq. 14) must be taken from
Table 2, we see that the three functions in any
column of the direction cosine matrix S transform like functions
of species
F_{2x},F_{2y},F_{2z}
as far as proper rotations are concerned. Since the matrices N in
(eq. 18) must also be taken from
Table 2, and since the transformation (eq. 18) involves −N,
not +N, the columns of the direction cosine matrix do not
transform like
F_{2x},F_{2y},F_{2z}
as far as improper rotations are concerned.
It happens, however, that the matrices in Table 3
are identical to those in Table 2 for proper rotations and equal to the
negatives of those in Table 2 for improper rotations. Thus, the three
functions in any column of the direction cosine matrix
S(χθφ)
transform like functions of species
F_{1x},F_{1y},F_{1z}
under the operations of the full molecular symmetry group T_{d}.