If only a purely mathematical familiarity is acquired, the formalism can
actually be rather easily misused. One need only observe that the point groups
*C*_{2h}, *C*_{2v}, and
*D*_{2} are isomorphic, for example, to conclude that all purely
group theoretical algebra must be identical for these three point groups.
Nevertheless, in spite of the fact that purely mathematical statements must be
identical, it is well known [2,4] that for molecules
of the point group *C*_{2h}, rotations and translations
(and therefore Raman and infrared intensity operators) *never* belong to
the same symmetry species, for *D*_{2} they *always* belong
to the same species in pairs, and for *C*_{2v} the
situation is mixed. These differences arise because the same group theoretical
formalism must be applied to the molecules in three rather different ways
depending on the physics of the situation.

One should thus resist the temptation to apply the formalism of vector coupling and recoupling developed for atoms to problems in methane without sufficient regard for the physical distinction between laboratory-fixed rotations and point-group rotations, or between the laboratory-fixed inversion operation and the molecule-fixed inversion operation. Such application can give rise to a sequence of absolutely correct mathematical operations followed by a false physical interpretation. (Note that difficulties of this nature do not arise in atomic problems, for which the formalism of angular momentum coupling was first developed and systematized, since atom-fixed coordinate axes are never considered.)

In the remainder of this section we shall discuss in general some of the ways in which irreducible tensor formalism can be applied to methane, without attempting to illustrate or to choose between the entire variety of notational and phase-factor conventions found in the group-theoretical literature and textbooks.

(eq. 58) |

There are further, not always obvious, modifications of this notation in the literature. The meaning of the quantity in (eq. 58) can be deduced almost immediately from the following equation

(eq. 59) |

From one point of view, (eq. 59) gives a relationship between
eigenfunctions |* j*_{1 }*m*_{1}⟩ of
the angular momentum operators **j**_{1}^{2},
* j*_{1z }, eigenfunctions
|* j*_{2 }*m*_{2}⟩ of
** j**_{2}^{2},
* j*_{2z }, and eigenfunctions
|* j*_{1}* j*_{2}* jm*⟩ of
** j**_{1}^{2},
** j**_{2}^{2},
** j**^{2} ≡
(**j**_{1} + **j**_{2})^{2},
* j _{z }*. From another point of view, (eq. 59)
gives a relationship between the (2