Understanding Ortho-hydrogen Properties

Ortho-hydrogen is a spin isomer of molecular hydrogen (H₂). Its properties are determined by the spin state of the two nuclei (protons) and the rotational state of the molecule.

Nuclear Spin Wavefunction Symmetry

Each proton in H₂ has a nuclear spin quantum number $I = 1/2$. The total nuclear spin ($I_{total}$) can be either 0 (singlet state) or 1 (triplet state).

• Ortho-hydrogen corresponds to the total nuclear spin $I_{total} = 1$. This state has a symmetric nuclear spin wavefunction. There are three such states (triplet).
• Para-hydrogen corresponds to the total nuclear spin $I_{total} = 0$. This state has an antisymmetric nuclear spin wavefunction (singlet).

Rotational Quantum Number and Overall Symmetry

For a homonuclear diatomic molecule like H₂, the total wavefunction must be antisymmetric with respect to the exchange of the identical nuclei. The total wavefunction is a product of electronic, vibrational, nuclear spin, and rotational wavefunctions.

• For ortho-hydrogen, the nuclear spin wavefunction is symmetric.
• To ensure the overall wavefunction is antisymmetric, the product of the electronic, vibrational, and rotational wavefunctions must be antisymmetric. For the ground electronic and vibrational states (which are symmetric), the rotational wavefunction must be antisymmetric.
• The rotational wavefunction's symmetry is determined by the rotational quantum number $J$. It is proportional to $ (\minus 1)^J $.
• An antisymmetric rotational wavefunction occurs when $J$ is odd ($(\minus 1)^J = \minus 1$).

Therefore, for ortho-hydrogen:

• The nuclear wavefunction is symmetric.
• The rotational quantum number ($J$) must be odd.

Conclusion

Ortho-hydrogen is characterized by a symmetric nuclear wavefunction and an odd rotational quantum number.