## Methane Symmetry Operations

#### 9.2   Nuclear spin statistics and overall parities

We now come to the necessity of imposing Fermi-Dirac or Bose-Einstein statistics, respectively, on the complete molecular wave functions of CH4 and CD4. Fermi-Dirac statistics require the complete wave function to be invariant with respect to those proton permutations which can be represented as the product of an even number of binary cycles, and require it to transform into its negative under permutations which can be represented as the product of an odd number of binary cycles. Bose-Einstein statistics require the complete wave function to be invariant with respect to both kinds of deuteron permutations.

Note that neither Fermi-Dirac nor Bose-Einstein statistics place requirements on the behavior of the complete wave function under the laboratory-fixed inversion operation. As a consequence we need consider complete wave function behavior only under the proper rotations in Table 1. Note further, that the C3 proper rotations and the C2 proper rotations in methane both correspond to an even number of binary permutation cycles, i.e., that no proper rotations in methane correspond to an odd number of binary permutation cycles.

We conclude from the above statements that complete wave functions of species A1 or A2 satisfy Fermi-Dirac statistics, while functions of species E, F1, and F2 do not. But, likewise, complete wave functions of species A1 or A2 satisfy Bose-Einstein statistics, while functions of species E, F1, and F2 do not.

One might ask what information has been lost by ignoring the sense-reversing point-group operations when dealing with nuclear spin statistics. In fact, what is lost is the correlation between the parity of a state and its group-theoretical symmetry species [23]. The σd planes, for example, represent permutation-inversion operations of the form (ij)*. A complete CH4 wave function should transform into its negative under the permutation (ij) because of Fermi-Dirac statistics. If such a change fails to take place under (ij)*, we must assume that the effects of the permutation (ij) were counteracted by the transformation induced by the laboratory-fixed inversion operation *. An examination of Table 1 using these arguments shows that complete wave functions for CH4 of species A1 must have negative parity, while those of species A2 must have positive parity. Complete wave functions for CD4 of species A1 must have positive parity, while those of species A2 have negative parity.

Table 17 indicates which nuclear spin functions must be combined with rovibrational functions of given symmetry to yield allowed overall wave-functions, and thus illustrates how the well-known [32] statistical weights arise. Table 17 can be constructed from Table 10, Table 15, and Table 16.

#### Table 17. Statistical weights for CH4 and CD4

Molecule Rovibrational
species
Nuclear spin
functions
Overall
species
Statistical
weight
CH4 A1 A1(I = 2) A1 (2I + 1) = 5
A2 A1(I = 2) A2 (2I + 1) = 5
E E(I = 0) A1 + A2 2(2I + 1) = 2
F1 F2(I = 1) A2 (2I + 1) = 3
F2 F2(I = 1) A1 (2I + 1) = 3
CD4 A1 A1(I = 4, 2, 0) A1 ΣI(2I + 1) = 15
A2 A1(I = 4, 2, 0) A2 ΣI(2I + 1) = 15
E E(I = 2, 0) A1 + A2 I(2I + 1) = 12
F1 F2(I = 3, 2, 1)
F1(I = 1)
A2
A1
ΣI(2I + 1) = 18
F2 F2(I = 3, 2, 1)
F1(I = 1)
A1
A2
ΣI(2I + 1) = 18