We now come to the necessity of imposing Fermi-Dirac or Bose-Einstein statistics, respectively, on the complete molecular wave functions of CH_{4} and CD_{4}. 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 C_{3} proper rotations and the C_{2} 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 A_{1} or A_{2} satisfy Fermi-Dirac statistics, while functions of species E, F_{1}, and F_{2} do not. But, likewise, complete wave functions of species A_{1} or A_{2} satisfy Bose-Einstein statistics, while functions of species E, F_{1}, and F_{2} 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 σ_{d} planes, for example, represent permutation-inversion operations of the form (ij)*. A complete CH_{4} 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 CH_{4} of species A_{1} must have negative parity, while those of species A_{2} must have positive parity. Complete wave functions for CD_{4} of species A_{1} must have positive parity, while those of species A_{2} 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.
Molecule | Rovibrational species |
Nuclear spin functions |
Overall species |
Statistical weight |
---|---|---|---|---|
CH_{4} | A_{1} | A_{1}(I = 2) | A_{1} | (2I + 1) = 5 |
A_{2} | A_{1}(I = 2) | A_{2} | (2I + 1) = 5 | |
E | E(I = 0) | A_{1} + A_{2} | 2(2I + 1) = 2 | |
F_{1} | F_{2}(I = 1) | A_{2} | (2I + 1) = 3 | |
F_{2} | F_{2}(I = 1) | A_{1} | (2I + 1) = 3 | |
CD_{4} | A_{1} | A_{1}(I = 4, 2, 0) | A_{1} | Σ_{I}(2I + 1) = 15 |
A_{2} | A_{1}(I = 4, 2, 0) | A_{2} | Σ_{I}(2I + 1) = 15 | |
E | E(I = 2, 0) | A_{1} + A_{2} | 2Σ_{I}(2I + 1) = 12 | |
F_{1} | F_{2}(I = 3, 2, 1) F_{1}(I = 1) |
A_{2} A_{1} |
Σ_{I}(2I + 1) = 18 | |
F_{2} | F_{2}(I = 3, 2, 1) F_{1}(I = 1) |
A_{1} A_{2} |
Σ_{I}(2I + 1) = 18 |