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 Jx , Jy , Jz transform as F1, we are interested in the species of F14. However, we need actually only consider species belonging to the symmetrized fourth power [F14]. Operators belonging to other species contained in F1 × F1 × F1 × F1 can be written in terms of products cubic and lower in the angular momentum components, by making use of the commutation relations
Table 9 indicates that [F14] contains the representation A1 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)
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 A1 for any values of DJ and Dt . We note also that the second term contains the operator J2Jz2, associated in a symmetric top with the constant DJK, and the operator Jz4, associated in a symmetric top with the constant DK.
Jx , Jy , Jz can also be considered to transform according to the irreducible representation Dg(1) of the continuous three-dimensional rotation-reflection group. The symmetrized fourth power [(Dg(1))4] contains the species Dg(0) + Dg(2) + Dg(4). It can be shown  that the first term in (eq. 63) transforms as Dg(0) under the three-dimensional rotation-reflection group, while the second term corresponds to a linear combination of terms transforming according to Dg(4). If we represent functions transforming according to Dg(4) by fm(4), where -4 ≤ m ≤ +4, the second term in (eq. 63) is proportional to 
It is common [10, 14, 65] to handle higher-order vibration-rotation operators for υs = 1 states, s = 3 or 4, rather formally as follows. Consider a set of (2k1 + 1) vibrational operators fm1(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 Qsx , Qsy , Qsz , their conjugate linear momenta, and the components Lsx , Lsy , Lsz of the vibrational angular momentum. Consider also a set of (2k2 + 1) rotational operators gm2(k2) which transform according to D(k2) and are taken to be functions of Jx , Jy , Jz . It is then possible to couple these operators using the vector coupling techniques to obtain vibration-rotation operators
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 Dg(0) under overall rotations in the laboratory, and according to A1 under the point-group symmetry operations. Loosely speaking, the laboratory-fixed "projections" of the rotational tensors must be correctly coupled to the laboratory-fixed "projections" of the spin tensors, and the molecule-fixed "projections" of the rotational tensors must be correctly coupled to the molecule-fixed "projections" of the spin tensors.
The first quantity on the right is a so-called reduced matrix element; its value does not depend on m1, m2, or m3 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), (2j1 + 1) (2j2 + 1) (2j3 + 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.