(eq. 38) |

where *Q _{s}*

(eq. 39) |

where *P*_{θs}
= −*i*(∂/∂θ_{s}),
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 *L _{sx }*,

(eq. 40) |

belong to the eigenvalues *L*(*L* + 1)^{2} and *k _{L}*, respectively, of

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
(θ_{s}φ_{s}), whereas rotational configuration space
is characterized by three (χθφ). The analog χ_{s} 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.

(eq. 41) |

By extension, we require *Q _{sx }*,

The question now arises of how the eigenfunctions
|*L k _{L}*⟩ 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

(eq. 42) |

In (eq. 42) we have introduced an extraneous variable χ_{s},
and generated with its help a complete 3 × 3 matrix of the form
*S*(χ_{s}θ_{s}φ_{s})
from the row vector [sinθ_{s} cosφ_{s},
sinθ_{s} sinφ_{s},
cosθ_{s}]. 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*(χ_{s}θ_{s}φ_{s})
transforms and this *third row* does not involve the extraneous variable
χ_{s}.

We now generalize (eq. 42) by omitting the row vector [0 0 1]
and the scalar *Q _{s}*, to obtain

(eq. 43) |

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 [22] of the matrices

(eq. 44) |

and extending, as in Sec. 8, validity of the
transformed (eq. 43) from *L* = 1 to all *L*, we find
that

(eq. 45) |

Specializing this to *k*″ = 0, and making use of
(eq. 40) we obtain

(eq. 46) |

Defining the eigenfunctions
| _{L}⟩ of ^{2}, _{z} to be

(eq. 47) |

where *L* = and
*k _{L}* = −

(eq. 48) |

an expression analogous to (eq. 33).

(eq. 49) |

Direct substitution of (θ_{s})_{new} and
(φ_{s})_{new} from (eq. 49) into Wigner's
[22]
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 [14] as
*D _{g}*