Consider first the form of the centrifugal distortion operator, which consists
of terms quartic in the molecule-fixed components of the total angular momentum
operator. Since *J _{x }*,

(eq. 62) |

Table 9 indicates that
[*F*_{1}^{4}] contains the representation
*A*_{1} only twice. Thus, there are for methane only two linearly
independent fourth power expressions in the total angular momentum components,
associated with two independent quartic centrifugal distortion constants. The
quartic centrifugal distortion operator is expressed in many different
notations in the literature. One which parallels symmetric top notation
somewhat is the following (suppressing 's)

(eq. 63) |

The construction of this operator from first principles is beyond the scope of
this work, but it can be verified using
(eq. 5) and
Table 3, that it is of species
*A*_{1} for any values of *D _{J}* and

*J _{x }*,

(eq. 64) |

It is common [10, 14,
65] to handle higher-order vibration-rotation
operators for *υ*_{s} = 1 states, *s* = 3
or 4, rather formally as follows. Consider a set of
(2*k*_{1} + 1) vibrational operators
*f*_{m1}^{(k1)} which
transform according to the irreducible representation
*D*^{(k1)} of the continuous three-dimensional
rotation group. These vibrational operators are taken to be functions of the
vibrational coordinates *Q _{sx }*,

(eq. 65) |

which transform according to *D*^{(k)}. Matrix elements of
these coupled vibration-rotation operators can be obtained formally using the
results of irreducible tensor theory.

The application of symmetry considerations to the resulting expressions, which
involve tensors in the rotational operators and tensors in the nuclear spin
operators, is particularly complicated because care must be taken to insure
that final coupled tensor operators transform as
*D _{g}*

(eq. 66) |

The first quantity on the right is a so-called reduced matrix element; its
value does not depend on *m*_{1}, *m*_{2}, or
*m*_{3} and must in general be calculated for the specific
molecular problem under consideration. The second factor is a vector coupling
coefficient; its value is determined by symmetry and can be tabulated once and
for all, or conveniently manipulated algebraically. Because of (eq. 66),
(2*j*_{1} + 1) (2*j*_{2} + 1)
(2*j*_{3} + 1) matrix elements can be related to a
single unknown parameter, achieving thereby an enormous simplification and
reduction of labor.

Equation (66) gives an example of a matrix element in an uncoupled basis set.
It is also written in terms of 3 - *j* symbols in the literature.
Extensive formalism has been developed to yield matrix elements of coupled
operators in coupled basis sets, which requires the introduction of
6 - *j* and 9 - *j* symbols, etc. There are
numerous different phase conventions used in the literature in the definitions
of 3 - *j* symbols and related quantities. Users of irreducible
tensor formalism must be absolutely certain that their own applications are
internally self-consistent.