Molecular spin statistics and identical nuclei

Identical nuclei of integer spin are composite bosons and those of half-integer spin are composite fermions, provided their internal nuclear states remain fixed. Exchanging the nuclei requires the full molecular wavefunction to be symmetric or antisymmetric accordingly. Electronic, vibrational, rotational, and nuclear-spin factors all contribute to that sign. For ground-state hydrogen and deuterium in the simple electronic and vibrational state considered here, the electronic-vibrational factor is symmetric, leaving rotation and spin to supply the required product.

Exchanging the two nuclei reverses the diatomic internuclear direction. A spherical harmonic of rotational quantum number J changes by $(-1)^J$, so even J is symmetric and odd J antisymmetric. This is a nuclear permutation statement; it should not be substituted indiscriminately for an electronic gerade/ungerade label. Other electronic states can contribute another exchange sign, and different isotopes such as H and D are distinguishable nuclei, so the same identical-pair restriction does not apply to HD.

Each nucleus has $2I+1$ spin-projection basis states. Two equal projections make symmetric product states. For each pair of unequal projections, the sum and difference of their two orderings make one symmetric and one antisymmetric spin function. This counting gives $(2I+1)(I+1)$ symmetric functions and $(2I+1)I$ antisymmetric functions, adding to the expected square of the one-nucleus state count. It concerns dimensions of exchange sectors rather than classical alignment probabilities.

For two protons, I is one half: the symmetric sector is the three-state triplet and the antisymmetric sector the one-state singlet. In ground electronic-vibrational H two, fermionic exchange requires even J to pair with the antisymmetric singlet, called para, and odd J with the symmetric triplet, called ortho. For two deuterons, I is one and the required total sign is bosonic; the symmetric six-state sector pairs with even J and the antisymmetric three-state sector with odd J under the same spatial assumption.

The three-to-one proton spin-state count does not mean the ortho-to-para population ratio is always three. Rotational energies and their degeneracies enter the partition sums, and at low temperature the lowest para rotational state is favored in equilibrium. A high-temperature limit can approach the familiar statistical ratio when many even and odd rotational states contribute comparably. Pairwise level ratios and complete spin-isomer fractions are different calculations, so a two-level estimate must not be presented as the full high-temperature partition sum.

Spin conversion can be slow because ordinary electric-dipole or Raman interactions often preserve nuclear spin to high accuracy. A cooled sample may therefore retain a nonequilibrium ortho/para composition until a suitable conversion mechanism acts. Spectral line alternation also depends on thermal populations and transition strengths in addition to spin weights. The symmetry rule defines which states are allowed; kinetics determines whether those allowed sectors equilibrate during an experiment. Neither the equilibrium rule nor a degeneracy factor by itself supplies a conversion timescale.