where Qs2 = Qsx2 + Qsy2 + Qsz2. The vibrational angular momentum operators given in (eq. 36) become, after a tedious but well known calculation,
where Pθs = −i(∂/∂θs), etc., and s = 3 or 4. Note that entries in the first column of the 3 × 3 matrix on the right of (eq. 39) are arbitrary, since the first component of the column vector on the right is zero.
Comparison of (eq. 25) and (eq. 39) shows that the form of the operators Lsx , Lsy , Lsz is identical to the form of the operators JX, JY, JZ provided that we replace the momentum pχ in (eq. 25) by zero. [Attempts to relabel φs as χs , and to write (eq. 39) in a fashion analogous to (eq. 23) fail because of difficulties with the minus signs.] We thus conclude, or prove by direct substitution, that eigenfunctions |L kL〉 of the form
belong to the eigenvalues L(L + 1)2 and kL, respectively, of L2 and Lz.
The normalization factors in (eq. 26) and (eq. 40) differ by (2π)1/2 because vibrational configuration space for a triply degenerate vibration is characterized by two angular variables (θsφs), whereas rotational configuration space is characterized by three (χθφ). The analog χs of the third Eulerian angle has been set to zero in (eq. 40). It can be given any value without altering (j)µ′ µ functions having µ′ = 0.
By extension, we require Qsx , Qsy , Qsz to transform according to this equation under any pure rotation.
The question now arises of how the eigenfunctions |L kL〉 transform. This can be determined rather easily by making use of some slightly formalistic arguments, designed to make the transformation algebra of this section analogous to that used in previous sections.
Equation (eq. 41) can be rewritten in terms of the spherical polar forms of the vibrational variables, and recast somewhat to read
In (eq. 42) we have introduced an extraneous variable χs, and generated with its help a complete 3 × 3 matrix of the form S(χsθsφs) from the row vector [sinθs cosφs, sinθs sinφs, cosθs]. Actually, however, because of the presence of the row vector [0 0 1], (eq. 42) makes a statement only about how the third row of the matrix S(χsθsφs) transforms and this third row does not involve the extraneous variable χs.
We now generalize (eq. 42) by omitting the row vector [0 0 1] and the scalar Qs, to obtain
It can be seen that this generalized equation introduces no inconsistencies with respect to the physically meaningful (eq. 42). Subjecting (eq. 43) to the similarity transformation defined by (eq. 30) and (eq. 31), making use of a property  of the matrices
and extending, as in Sec. 8, validity of the transformed (eq. 43) from L = 1 to all L, we find that
Specializing this to k″ = 0, and making use of (eq. 40) we obtain
Defining the eigenfunctions | L〉 of 2, z to be
where L = and kL = −L insures that matrix elements of the "reversed" ladder operators x iy are real and positive. Equation (46) can then be rewritten, using the | −k〉 functions rather than the |L +k〉, as
an expression analogous to (eq. 33).
Direct substitution of (θs)new and (φs)new from (eq. 49) into Wigner's  functions as used in (eq. 40) shows that the vibrational wave functions | L〉 transform into (−1)L times themselves under the molecule-fixed inversion operation i, i.e., that the vibrational wave functions | L〉 for a triply-degenerate mode transform  as Dg(L) for even L and as Du(L) for odd L.