Longuet-Higgins [1] has pointed out that an alternative
and more powerful way of treating molecular symmetry properties begins not with
the traditional concepts of crystallographic point groups, but rather with
concepts associated with the exchange of identical particles. Since
CH_{4} has four identical protons, there are 24 possible exchanges of
identical particles. These permutations taken together with the
laboratory-fixed inversion of the coordinates of all particles give rise to a
group containing 48 operations.

Longuet-Higgins [1] has further pointed out that only feasible permutation-inversion operations need be included in the molecular symmetry group, since only they lead to physically meaningful information. Qualitatively speaking, a feasible operation is one which the molecule can actually carry out physically within the time scale associated with the resolution of the experiment.

Now CH_{4} has two and only two non-superposible equilibrium frameworks,
the one shown in Fig. 1 and another formed, say,
by exchanging hydrogens 3 and 4. It is believed that transitions between
these two non-superposible frameworks in CH_{4} take place at a rate
much slower than 1 cycle s^{−1}, giving rise to spectral
splittings much smaller than 10^{−10} cm^{−1}. For the
purposes of this article we shall consider as feasible only the
24 permutation-inversion operations which do not change one of the
non-superposible frameworks into the other. (For some discussion of the group
theory when all 48 permutation-inversion operations are included in the
molecular symmetry group, see Ref. [23].)

The feasible permutation-inversion operations form a group isomorphic with
*T _{d}*. In the notation of permutation cycles, they are:

A convention involved in applying these permutation-inversion operations is
fixed by defining the effect of (123) *on a function of laboratory-fixed
coordinates and momenta associated with individual particles* to be that of
everywhere substituting the coordinates and momenta of particle 2 for
those of particle 1, those of particle 3 for those of
particle 2, and those of particle 1 for those of particle 3. In
equation form

(eq. 8) |

Note that the subscripts in (eq. 8) transform just oppositely from the
numerical labels in Fig. 1, e.g., since the
coordinates of particle 1 are replaced by coordinates of particle 2,
the numeral 2 in Fig. 1 is replaced by the numeral 1. It is
particularly easy to fall into confusion, and indeed error, if this subtle
distinction is not scrupulously respected. Indeed, the correct result for a
product of two permutation-inversion operations must be determined by applying
algebraic transformations to a function
*f* (**R**_{1}, **R**_{2},
**R**_{3}, **R**_{4}), rather than by
applying geometric transformations to diagrams like Fig. 1 and those
appearing later.