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Methane Symmetry Operations - Permutation-Inversion

3.   Permutation-Inversion Group Ideas

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 CH4 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 CH4 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 CH4 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 Td. In the notation of permutation cycles, they are: E; (132), (123), (134), (143), (124), (142), (243), (234); (13)(24), (14)(23), (12)(34); (1432)*, (1234)*, (1342)*, (1243)*, (1324)*, (1423)*; and (24)*, (13)*, (14)*, (23)*, (34)*, (12)*. Following Longuet-Higgins a * indicates the laboratory-fixed inversion operation.

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

$$\left[\begin{array}{c} (\mbox{$R$}_1)_{\rm new} \\ (\mbox{$R$}_2)_{\rm new} \\ (\mbox{$R$}_3)_{\rm new} \end{array}\right] = (123) \left[\begin{array}{c} (\mbox{$R$}_1)\\ ~\\ (\mbox{$R$}_2)\\ ~\\ (\mbox{$R$}_3) \end{array}\right] = \left[\begin{array}{c} (\mbox{$R$}_2)\\ ~\\ (\mbox{$R$}_3)\\ ~\\ (\mbox{$R$}_1) \end{array}\right] ~ .$$

(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 (R1, R2, R3, R4), rather than by applying geometric transformations to diagrams like Fig. 1 and those appearing later.


Created September 20, 2016, Updated June 2, 2021